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Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月19日(火)07時49分17秒
返信・引用
  (%i1) solve(p^2+p+41=k*81,p);
                  sqrt(324 k - 163) + 1      sqrt(324 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(324*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        81
において、
(%i6) a:0;
(%o6)                                  0
(%i7) mod(a^2+a,81);
(%o7)                                  0
(%i8) a:1;
(%o8)                                  1
(%i9) mod(a^2+a,81);
(%o9)                                  2
(%i10) a:2;
(%o10)                                 2
(%i11) mod(a^2+a,81);
(%o11)                                 6
(%i12) a:3;
(%o12)                                 3
(%i13) mod(a^2+a,81);
(%o13)                                12
(%i14) a:4;
(%o14)                                 4
(%i15) mod(a^2+a,81);
(%o15)                                20
(%i16) a:5;
(%o16)                                 5
(%i17) mod(a^2+a,81);
(%o17)                                30
(%i18) a:6;
(%o18)                                 6
(%i19) mod(a^2+a,81);
(%o19)                                42
(%i20) a:7;
(%o20)                                 7
(%i21) mod(a^2+a,81);
(%o21)                                56
(%i22) a:8;
(%o22)                                 8
(%i23) mod(a^2+a,81);
(%o23)                                72
(%i24) a:9;
(%o24)                                 9
(%i25) mod(a^2+a,81);
(%o25)                                 9
(%i26) a:10;
(%o26)                                10
(%i27) mod(a^2+a,81);
(%o27)                                29
(%i28) a:11;
(%o28)                                11
(%i29) mod(a^2+a,81);
(%o29)                                51
(%i30) a:12;
(%o30)                                12
(%i31) mod(a^2+a,81);
(%o31)                                75
(%i32) a:13;
(%o32)                                13
(%i33) mod(a^2+a,81);
(%o33)                                20
(%i34) a:14;
(%o34)                                14
(%i35) mod(a^2+a,81);
(%o35)                                48
(%i36) a:15;
(%o36)                                15
(%i37) mod(a^2+a,81);
(%o37)                                78
(%i38) a:16;
(%o38)                                16
(%i39) mod(a^2+a,81);
(%o39)                                29
(%i40) a:17;
(%o40)                                17
(%i41) mod(a^2+a,81);
(%o41)                                63
(%i42) a:18;
(%o42)                                18
(%i43) mod(a^2+a,81);
(%o43)                                18
(%i44) a:19;
(%o44)                                19
(%i45) mod(a^2+a,81);
(%o45)                                56
(%i46) a:20;
(%o46)                                20
(%i47) mod(a^2+a,81);
(%o47)                                15
(%i48) a:21;
(%o48)                                21
(%i49) mod(a^2+a,81);
(%o49)                                57
(%i50) a:22;
(%o50)                                22
(%i51) mod(a^2+a,81);
(%o51)                                20
(%i52) a:23;
(%o52)                                23
(%i53) mod(a^2+a,81);
(%o53)                                66
(%i54) a:24;
(%o54)                                24
(%i55) mod(a^2+a,81);
(%o55)                                33
(%i56) a:25;
(%o56)                                25
(%i57) mod(a^2+a,81);
(%o57)                                 2
(%i58) a:26;
(%o58)                                26
(%i59) mod(a^2+a,81);
(%o59)                                54
(%i60) a:27;
(%o60)                                27
(%i61) mod(a^2+a,81);
(%o61)                                27
(%i62) a:28;
(%o62)                                28
(%i63) mod(a^2+a,81);
(%o63)                                 2
(%i64) a:29;
(%o64)                                29
(%i65) mod(a^2+a,81);
(%o65)                                60
(%i66) a:30;
(%o66)                                30
(%i67) mod(a^2+a,81);
(%o67)                                39
(%i68) a:31;
(%o68)                                31
(%i69) mod(a^2+a,81);
(%o69)                                20
(%i70) a:32;
(%o70)                                32
(%i71) mod(a^2+a,81);
(%o71)                                 3
(%i72) a:33;
(%o72)                                33
(%i73) mod(a^2+a,81);
(%o73)                                69
(%i74) a:34;
(%o74)                                34
(%i75) mod(a^2+a,81);
(%o75)                                56
(%i76) a:35;
(%o76)                                35
(%i77) mod(a^2+a,81);
(%o77)                                45
(%i78) a:36;
(%o78)                                36
(%i79) mod(a^2+a,81);
(%o79)                                36
(%i80) a:37;
(%o80)                                37
(%i81) mod(a^2+a,81);
(%o81)                                29
(%i82) a:38;
(%o82)                                38
(%i83) mod(a^2+a,81);
(%o83)                                24
(%i84) a:39;
(%o84)                                39
(%i85) mod(a^2+a,81);
(%o85)                                21
(%i86) a:40;
(%o86)                                40
(%i87) mod(a^2+a,81);
(%o87)                                20
(%i88) a:41;
(%o88)                                41
(%i89) mod(a^2+a,81);
(%o89)                                21
mod(41,81)=41より、mod(x(x+1),81)=40でなければならない。しかし、mod(x(x+1),81)≠40であり、kは整数にならない。
 
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月18日(月)08時04分55秒
返信・引用
  (%i1) solve(p^2+p+41=k*79,p);
                  sqrt(316 k - 163) + 1      sqrt(316 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(316*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        79
において、
(%i4) a:0;
(%o4)                                  0
(%i5) mod(a^2+a,79);
(%o5)                                  0
(%i6) a:1;
(%o6)                                  1
(%i7) mod(a^2+a,79);
(%o7)                                  2
(%i8) a:2;
(%o8)                                  2
(%i9) mod(a^2+a,79);
(%o9)                                  6
(%i10) a:3;
(%o10)                                 3
(%i11) mod(a^2+a,79);
(%o11)                                12
(%i12) a:4;
(%o12)                                 4
(%i13) mod(a^2+a,79);
(%o13)                                20
(%i14) a:5;
(%o14)                                 5
(%i15) mod(a^2+a,79);
(%o15)                                30
(%i16) a:6;
(%o16)                                 6
(%i17) mod(a^2+a,79);
(%o17)                                42
(%i18) a:7;
(%o18)                                 7
(%i19) mod(a^2+a,79);
(%o19)                                56
(%i20) a:8;
(%o20)                                 8
(%i21) mod(a^2+a,79);
(%o21)                                72
(%i22) a:9;
(%o22)                                 9
(%i23) mod(a^2+a,79);
(%o23)                                11
(%i24) a:10;
(%o24)                                10
(%i25) mod(a^2+a,79);
(%o25)                                31
(%i26) a:11;
(%o26)                                11
(%i27) mod(a^2+a,79);
(%o27)                                53
(%i28) a:12;
(%o28)                                12
(%i29) mod(a^2+a,79);
(%o29)                                77
(%i30) a:13;
(%o30)                                13
(%i31) mod(a^2+a,79);
(%o31)                                24
(%i32) a:14;
(%o32)                                14
(%i33) mod(a^2+a,79);
(%o33)                                52
(%i34) a:15;
(%o34)                                15
(%i35) mod(a^2+a,79);
(%o35)                                 3
(%i36) a:16;
(%o36)                                16
(%i37) mod(a^2+a,79);
(%o37)                                35
(%i38) a:17;
(%o38)                                17
(%i39) mod(a^2+a,79);
(%o39)                                69
(%i40) a:18;
(%o40)                                18
(%i41) mod(a^2+a,79);
(%o41)                                26
(%i42) a:19;
(%o42)                                19
(%i43) mod(a^2+a,79);
(%o43)                                64
(%i44) a:20;
(%o44)                                20
(%i45) mod(a^2+a,79);
(%o45)                                25
(%i46) a:21;
(%o46)                                21
(%i47) mod(a^2+a,79);
(%o47)                                67
(%i48) a:22;
(%o48)                                22
(%i49) mod(a^2+a,79);
(%o49)                                32
(%i50) a:23;
(%o50)                                23
(%i51) mod(a^2+a,79);
(%o51)                                78
(%i52) a:24;
(%o52)                                24
(%i53) mod(a^2+a,79);
(%o53)                                47
(%i54) a:25;
(%o54)                                25
(%i55) mod(a^2+a,79);
(%o55)                                18
(%i56) a:26;
(%o56)                                26
(%i57) mod(a^2+a,79);
(%o57)                                70
(%i58) a:27;
(%o58)                                27
(%i59) mod(a^2+a,79);
(%o59)                                45
(%i60) a:28;
(%o60)                                28
(%i61) mod(a^2+a,79);
(%o61)                                22
(%i62) a:29;
(%o62)                                29
(%i63) mod(a^2+a,79);
(%o63)                                 1
(%i64) a:30;
(%o64)                                30
(%i65) mod(a^2+a,79);
(%o65)                                61
(%i66) a:31;
(%o66)                                31
(%i67) mod(a^2+a,79);
(%o67)                                44
(%i68) a:32;
(%o68)                                32
(%i69) mod(a^2+a,79);
(%o69)                                29
(%i70) a:33;
(%o70)                                33
(%i71) mod(a^2+a,79);
(%o71)                                16
(%i72) a:34;
(%o72)                                34
(%i73) mod(a^2+a,79);
(%o73)                                 5
(%i74) a:35;
(%o74)                                35
(%i75) mod(a^2+a,79);
(%o75)                                75
(%i76) a:36;
(%o76)                                36
(%i77) mod(a^2+a,79);
(%o77)                                68
(%i78) a:37;
(%o78)                                37
(%i79) mod(a^2+a,79);
(%o79)                                63
(%i80) a:38;
(%o80)                                38
(%i81) mod(a^2+a,79);
(%o81)                                60
(%i82) a:38;
(%o82)                                38
(%i83) mod(a^2+a,79);
(%o83)                                60
(%i84) a:39;
(%o84)                                39
(%i85) mod(a^2+a,79);
(%o85)                                59
(%i86) a:40;
(%o86)                                40
(%i87) mod(a^2+a,79);
(%o87)                                60
mod(41,79)=41より、mod(x(x+1),79)=38でなければならない。しかし、mod(x(x+1),79)≠38であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月17日(日)03時49分55秒
返信・引用
  (%i1) solve(p^2+p+41=k*77,p);
                  sqrt(308 k - 163) + 1      sqrt(308 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(308*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        77
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,77);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,77);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,77);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,77);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,77);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,77);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,77);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,77);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,77);
(%o20)                                72
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,77);
(%o22)                                13
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,77);
(%o24)                                33
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,77);
(%o26)                                55
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,77);
(%o28)                                 2
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,77);
(%o30)                                28
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,77);
(%o32)                                56
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,77);
(%o34)                                 9
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,77);
(%o36)                                41
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,77);
(%o38)                                75
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,77);
(%o40)                                34
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,77);
(%o42)                                72
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,77);
(%o44)                                35
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,77);
(%o46)                                 0
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,77);
(%o48)                                44
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,77);
(%o50)                                13
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,77);
(%o52)                                61
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,77);
(%o54)                                34
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,77);
(%o56)                                 9
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,77);
(%o58)                                63
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,77);
(%o60)                                42
(%i61) a:29;
(%o61)                                29
(%i62) mod(a^2+a,77);
(%o62)                                23
(%i63) a:30;
(%o63)                                30
(%i64) mod(a^2+a,77);
(%o64)                                 6
(%i65) a:31;
(%o65)                                31
(%i66) mod(a^2+a,77);
(%o66)                                68
(%i67) a:32;
(%o67)                                32
(%i68) mod(a^2+a,77);
(%o68)                                55
(%i69) a:33;
(%o69)                                33
(%i70) mod(a^2+a,77);
(%o70)                                44
(%i71) a:34;
(%o71)                                34
(%i72) mod(a^2+a,77);
(%o72)                                35
(%i73) a:35;
(%o73)                                35
(%i74) mod(a^2+a,77);
(%o74)                                28
(%i75) a:36;
(%o75)                                36
(%i76) mod(a^2+a,77);
(%o76)                                23
(%i77) a:37;
(%o77)                                37
(%i78) mod(a^2+a,77);
(%o78)                                20
(%i79) a:38;
(%o79)                                38
(%i80) mod(a^2+a,77);
(%o80)                                19
(%i81) a:39;
(%o81)                                39
(%i82) mod(a^2+a,77);
(%o82)                                20
mod(41,77)=41より、mod(x(x+1),77)=36でなければならない。しかし、mod(x(x+1),77)≠36であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月15日(金)07時31分3秒
返信・引用
  (%i1) solve(p^2+p+41=k*75,p);
                  sqrt(300 k - 163) + 1      sqrt(300 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(300*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        75
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,75);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,75);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,75);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,75);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,75);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,75);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,75);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,75);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,75);
(%o20)                                72
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,75);
(%o22)                                15
(%i24) a:10;
(%o24)                                10
(%i25) mod(a^2+a,75);
(%o25)                                35
(%i26) a:11;
(%o26)                                11
(%i27) mod(a^2+a,75);
(%o27)                                57
(%i28) a:12;
(%o28)                                12
(%i29) mod(a^2+a,75);
(%o29)                                 6
(%i30) a:13;
(%o30)                                13
(%i31) mod(a^2+a,75);
(%o31)                                32
(%i32) a:14;
(%o32)                                14
(%i33) mod(a^2+a,75);
(%o33)                                60
(%i34) a:15;
(%o34)                                15
(%i35) mod(a^2+a,75);
(%o35)                                15
(%i36) a:16;
(%o36)                                16
(%i37) mod(a^2+a,75);
(%o37)                                47
(%i38) a:17;
(%o38)                                17
(%i39) mod(a^2+a,75);
(%o39)                                 6
(%i40) a:18;
(%o40)                                18
(%i41) mod(a^2+a,75);
(%o41)                                42
(%i42) a:19;
(%o42)                                19
(%i43) mod(a^2+a,75);
(%o43)                                 5
(%i44) a:20;
(%o44)                                20
(%i45) mod(a^2+a,75);
(%o45)                                45
(%i46) a:21;
(%o46)                                21
(%i47) mod(a^2+a,75);
(%o47)                                12
(%i48) a:22;
(%o48)                                22
(%i49) mod(a^2+a,75);
(%o49)                                56
(%i50) a:23;
(%o50)                                23
(%i51) mod(a^2+a,75);
(%o51)                                27
(%i52) a:24;
(%o52)                                24
(%i53) mod(a^2+a,75);
(%o53)                                 0
(%i54) a:25;
(%o54)                                25
(%i55) mod(a^2+a,75);
(%o55)                                50
(%i56) a:26;
(%o56)                                26
(%i57) mod(a^2+a,75);
(%o57)                                27
(%i58) a:27;
(%o58)                                27
(%i59) mod(a^2+a,75);
(%o59)                                 6
(%i60) a:28;
(%o60)                                28
(%i61) mod(a^2+a,75);
(%o61)                                62
(%i62) a:29;
(%o62)                                29
(%i63) mod(a^2+a,75);
(%o63)                                45
(%i64) a:30;
(%o64)                                30
(%i65) mod(a^2+a,75);
(%o65)                                30
(%i66) a:31;
(%o66)                                31
(%i67) mod(a^2+a,75);
(%o67)                                17
(%i68) a:32;
(%o68)                                32
(%i69) mod(a^2+a,75);
(%o69)                                 6
(%i70) a:33;
(%o70)                                33
(%i71) mod(a^2+a,75);
(%o71)                                72
(%i72) a:34;
(%o72)                                34
(%i73) mod(a^2+a,75);
(%o73)                                65
(%i74) a:35;
(%o74)                                35
(%i75) mod(a^2+a,75);
(%o75)                                60
(%i76) a:36;
(%o76)                                36
(%i77) mod(a^2+a,75);
(%o77)                                57
(%i78) a:37;
(%o78)                                37
(%i79) mod(a^2+a,75);
(%o79)                                56
(%i80) a:38;
(%o80)                                38
(%i81) mod(a^2+a,75);
(%o81)                                57
mod(41,75)=41より、mod(x(x+1),75)=34でなければならない。しかし、mod(x(x+1),75)≠34であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月14日(木)15時40分13秒
返信・引用
  (%i1) solve(p^2+p+41=k*73,p);
                  sqrt(292 k - 163) + 1      sqrt(292 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(292*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        73
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,73);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,73);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,73);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,73);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,73);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,73);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,73);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,73);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,73);
(%o20)                                72
(%i30) a:9;
(%o30)                                 9
(%i31) mod(a^2+a,73);
(%o31)                                17
(%i25) a:10;
(%o25)                                10
(%i26) mod(a^2+a,73);
(%o26)                                37
(%i27) a:11;
(%o27)                                11
(%i28) mod(a^2+a,73);
(%o28)                                59
(%i29) a:12;
(%o29)                                12
(%i83) mod(a^2+a,73);
(%o83)                                10
(%i32) a:13;
(%o32)                                13
(%i33) mod(a^2+a,73);
(%o33)                                36
(%i34) a:14;
(%o34)                                14
(%i35) mod(a^2+a,73);
(%o35)                                64
(%i36) a:15;
(%o36)                                15
(%i37) mod(a^2+a,73);
(%o37)                                21
(%i38) a:16;
(%o38)                                16
(%i39) mod(a^2+a,73);
(%o39)                                53
(%i40) a:17;
(%o40)                                17
(%i41) mod(a^2+a,73);
(%o41)                                14
(%i42) a:18;
(%o42)                                18
(%i43) mod(a^2+a,73);
(%o43)                                50
(%i44) a:19;
(%o44)                                19
(%i45) mod(a^2+a,73);
(%o45)                                15
(%i46) a:20;
(%o46)                                20
(%i47) mod(a^2+a,73);
(%o47)                                55
(%i48) a:21;
(%o48)                                21
(%i49) mod(a^2+a,73);
(%o49)                                24
(%i50) a:22;
(%o50)                                22
(%i51) mod(a^2+a,73);
(%o51)                                68
(%i52) a:23;
(%o52)                                23
(%i53) mod(a^2+a,73);
(%o53)                                41
(%i54) a:24;
(%o54)                                24
(%i55) mod(a^2+a,73);
(%o55)                                16
(%i56) a:25;
(%o56)                                25
(%i57) mod(a^2+a,73);
(%o57)                                66
(%i58) a:26;
(%o58)                                26
(%i59) mod(a^2+a,73);
(%o59)                                45
(%i60) a:27;
(%o60)                                27
(%i61) mod(a^2+a,73);
(%o61)                                26
(%i62) a:28;
(%o62)                                28
(%i63) mod(a^2+a,73);
(%o63)                                 9
(%i64) a:29;
(%o64)                                29
(%i65) mod(a^2+a,73);
(%o65)                                67
(%i66) a:30;
(%o66)                                30
(%i67) mod(a^2+a,73);
(%o67)                                54
(%i68) a:31;
(%o68)                                31
(%i69) mod(a^2+a,73);
(%o69)                                43
(%i70) a:32;
(%o70)                                32
(%i71) mod(a^2+a,73);
(%o71)                                34
(%i72) a:33;
(%o72)                                33
(%i73) mod(a^2+a,73);
(%o73)                                27
(%i74) a:34;
(%o74)                                34
(%i75) mod(a^2+a,73);
(%o75)                                22
(%i76) a:35;
(%o76)                                35
(%i77) mod(a^2+a,73);
(%o77)                                19
(%i78) a:36;
(%o78)                                36
(%i79) mod(a^2+a,73);
(%o79)                                18
(%i80) a:37;
(%o80)                                37
(%i81) mod(a^2+a,73);
(%o81)                                19
mod(41,73)=41より、mod(x(x+1),73)=32でなければならない。しかし、mod(x(x+1),73)≠32であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月13日(水)07時58分3秒
返信・引用
  (%i1) solve(p^2+p+41=k*71,p);
                  sqrt(284 k - 163) + 1      sqrt(284 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(284*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        71
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,71);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,71);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,71);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,71);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,71);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,71);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,71);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,71);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,71);
(%o20)                                 1
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,71);
(%o22)                                19
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,71);
(%o24)                                39
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,71);
(%o26)                                61
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,71);
(%o28)                                14
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,71);
(%o30)                                40
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,71);
(%o32)                                68
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,71);
(%o34)                                27
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,71);
(%o36)                                59
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,71);
(%o38)                                22
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,71);
(%o40)                                58
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,71);
(%o42)                                25
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,71);
(%o44)                                65
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,71);
(%o46)                                36
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,71);
(%o48)                                 9
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,71);
(%o50)                                55
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,71);
(%o52)                                32
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,71);
(%o54)                                11
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,71);
(%o56)                                63
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,71);
(%o58)                                46
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,71);
(%o60)                                31
(%i61) a:29;
(%o61)                                29
(%i62) mod(a^2+a,71);
(%o62)                                18
(%i63) a:30;
(%o63)                                30
(%i64) mod(a^2+a,71);
(%o64)                                 7
(%i65) a:31;
(%o65)                                31
(%i66) mod(a^2+a,71);
(%o66)                                69
(%i67) a:32;
(%o67)                                32
(%i68) mod(a^2+a,71);
(%o68)                                62
(%i69) a:33;
(%o69)                                33
(%i70) mod(a^2+a,71);
(%o70)                                57
(%i71) a:34;
(%o71)                                34
(%i72) mod(a^2+a,71);
(%o72)                                54
(%i73) a:35;
(%o73)                                35
(%i74) mod(a^2+a,71);
(%o74)                                53
(%i75) a:36;
(%o75)                                36
(%i76) mod(a^2+a,71);
(%o76)                                54
mod(41,71)=41より、mod(x(x+1),71)=30でなければならない。mod(x(x+1),71)=30があり、kは整数になる場合がある。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月12日(火)07時57分10秒
返信・引用 編集済
  (%i1) solve(p^2+p+41=k*69,p);
                  sqrt(276 k - 163) + 1      sqrt(276 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(276*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        69
において、
(%i3) a:0;
(%o3)                                  0
(%i5) mod(a^2+a,69);
(%o5)                                  0
(%i6) a:1;
(%o6)                                  1
(%i7) mod(a^2+a,69);
(%o7)                                  2
(%i8) a:2;
(%o8)                                  2
(%i9) mod(a^2+a,69);
(%o9)                                  6
(%i10) a:3;
(%o10)                                 3
(%i11) mod(a^2+a,69);
(%o11)                                12
(%i12) a:4;
(%o12)                                 4
(%i13) mod(a^2+a,69);
(%o13)                                20
(%i14) a:5;
(%o14)                                 5
(%i15) mod(a^2+a,69);
(%o15)                                30
(%i16) a:6;
(%o16)                                 6
(%i17) mod(a^2+a,69);
(%o17)                                42
(%i18) a:7;
(%o18)                                 7
(%i19) mod(a^2+a,69);
(%o19)                                56
(%i20) a:8;
(%o20)                                 8
(%i21) mod(a^2+a,69);
(%o21)                                 3
(%i22) a:9;
(%o22)                                 9
(%i23) mod(a^2+a,69);
(%o23)                                21
(%i24) a:10;
(%o24)                                10
(%i25) mod(a^2+a,69);
(%o25)                                41
(%i26) a:11;
(%o26)                                11
(%i27) mod(a^2+a,69);
(%o27)                                63
(%i28) a:12;
(%o28)                                12
(%i29) mod(a^2+a,69);
(%o29)                                18
(%i30) a:13;
(%o30)                                13
(%i31) mod(a^2+a,69);
(%o31)                                44
(%i32) a:14;
(%o32)                                14
(%i33) mod(a^2+a,69);
(%o33)                                 3
(%i34) a:15;
(%o34)                                15
(%i35) mod(a^2+a,69);
(%o35)                                33
(%i36) a:16;
(%o36)                                16
(%i37) mod(a^2+a,69);
(%o37)                                65
(%i38) a:17;
(%o38)                                17
(%i39) mod(a^2+a,69);
(%o39)                                30
(%i40) a:18;
(%o40)                                18
(%i41) mod(a^2+a,69);
(%o41)                                66
(%i42) a:19;
(%o42)                                19
(%i43) mod(a^2+a,69);
(%o43)                                35
(%i44) a:20;
(%o44)                                20
(%i45) mod(a^2+a,69);
(%o45)                                 6
(%i46) a:21;
(%o46)                                21
(%i47) mod(a^2+a,69);
(%o47)                                48
(%i48) a:22;
(%o48)                                22
(%i49) mod(a^2+a,69);
(%o49)                                23
(%i50) a:23;
(%o50)                                23
(%i51) mod(a^2+a,69);
(%o51)                                 0
(%i52) a:24;
(%o52)                                24
(%i53) mod(a^2+a,69);
(%o53)                                48
(%i54) a:25;
(%o54)                                25
(%i55) mod(a^2+a,69);
(%o55)                                29
(%i56) a:26;
(%o56)                                26
(%i57) mod(a^2+a,69);
(%o57)                                12
(%i58) a:27;
(%o58)                                27
(%i59) mod(a^2+a,69);
(%o59)                                66
(%i60) a:28;
(%o60)                                28
(%i61) mod(a^2+a,69);
(%o61)                                53
(%i62) a:29;
(%o62)                                29
(%i63) mod(a^2+a,69);
(%o63)                                42
(%i64) a:30;
(%o64)                                30
(%i65) mod(a^2+a,69);
(%o65)                                33
(%i66) a:31;
(%o66)                                31
(%i67) mod(a^2+a,69);
(%o67)                                26
(%i68) a:32;
(%o68)                                32
(%i69) mod(a^2+a,69);
(%o69)                                21
(%i70) a:33;
(%o70)                                33
(%i71) mod(a^2+a,69);
(%o71)                                18
(%i72) a:34;
(%o72)                                34
(%i73) mod(a^2+a,69);
(%o73)                                17
(%i74) a:35;
(%o74)                                35
(%i75) mod(a^2+a,69);
(%o75)                                18
mod(41,69)=41より、mod(x(x+1),69)=28でなければならない。しかし、mod(x(x+1),69)≠28であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月10日(日)19時01分9秒
返信・引用
  (%i1) solve(p^2+p+41=k*67,p);
                  sqrt(268 k - 163) + 1      sqrt(268 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(268*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        67
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,67);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,67);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,67);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,67);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,67);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,67);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,67);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,67);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,67);
(%o20)                                 5
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,67);
(%o22)                                23
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,67);
(%o24)                                43
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,67);
(%o26)                                65
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,67);
(%o28)                                22
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,67);
(%o30)                                48
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,67);
(%o32)                                 9
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,67);
(%o34)                                39
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,67);
(%o36)                                 4
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,67);
(%o38)                                38
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,67);
(%o40)                                 7
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,67);
(%o42)                                45
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,67);
(%o44)                                18
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,67);
(%o46)                                60
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,67);
(%o48)                                37
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,67);
(%o50)                                16
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,67);
(%o52)                                64
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,67);
(%o54)                                47
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,67);
(%o56)                                32
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,67);
(%o58)                                19
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,67);
(%o60)                                 8
(%i61) a:29;
(%o61)                                29
(%i62) mod(a^2+a,67);
(%o62)                                66
(%i63) a:30;
(%o63)                                30
(%i64) mod(a^2+a,67);
(%o64)                                59
(%i65) a:31;
(%o65)                                31
(%i66) mod(a^2+a,67);
(%o66)                                54
(%i67) a:32;
(%o67)                                32
(%i68) mod(a^2+a,67);
(%o68)                                51
(%i69) a:33;
(%o69)                                33
(%i70) mod(a^2+a,67);
(%o70)                                50
(%i71) a:34;
(%o71)                                34
(%i72) mod(a^2+a,67);
(%o72)                                51
mod(41,67)=41より、mod(x(x+1),67)=26でなければならない。しかし、mod(x(x+1),67)≠26であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 9日(土)12時20分8秒
返信・引用
  (%i1) solve(p^2+p+41=k*65,p);
                  sqrt(260 k - 163) + 1      sqrt(260 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(260*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        65
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,65);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,65);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,65);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,65);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,65);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,65);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,65);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,65);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,65);
(%o20)                                 7
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,65);
(%o22)                                25
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,65);
(%o24)                                45
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,65);
(%o26)                                 2
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,65);
(%o28)                                26
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,65);
(%o30)                                52
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,65);
(%o32)                                15
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,65);
(%o34)                                45
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,65);
(%o36)                                12
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,65);
(%o38)                                46
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,65);
(%o40)                                17
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,65);
(%o42)                                55
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,65);
(%o44)                                30
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,65);
(%o46)                                 7
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,65);
(%o48)                                51
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,65);
(%o50)                                32
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,65);
(%o52)                                15
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,65);
(%o54)                                 0
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,65);
(%o56)                                52
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,65);
(%o58)                                41
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,65);
(%o60)                                32
(%i61) a:29;
(%o61)                                29
(%i62) mod(a^2+a,65);
(%o62)                                25
(%i63) a:30;
(%o63)                                30
(%i64) mod(a^2+a,65);
(%o64)                                20
(%i65) a:31;
(%o65)                                31
(%i66) mod(a^2+a,65);
(%o66)                                17
(%i67) a:32;
(%o67)                                32
(%i68) mod(a^2+a,65);
(%o68)                                16
(%i69) a:33;
(%o69)                                33
(%i70) mod(a^2+a,65);
(%o70)                                17
mod(41,65)=41より、mod(x(x+1),65)=24でなければならない。しかし、mod(x(x+1),65)≠24であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 8日(金)19時49分59秒
返信・引用
  (%i1) solve(p^2+p+41=k*63,p);
                  sqrt(252 k - 163) + 1      sqrt(252 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(252*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        63
において、
(%i3) a:0;
(%o3)                                  0
(%i5) mod(a^2+a,63);
(%o5)                                  0
(%i6) a:1;
(%o6)                                  1
(%i7) mod(a^2+a,63);
(%o7)                                  2
(%i8) a:2;
(%o8)                                  2
(%i9) mod(a^2+a,63);
(%o9)                                  6
(%i10) a:3;
(%o10)                                 3
(%i11) mod(a^2+a,63);
(%o11)                                12
(%i12) a:4;
(%o12)                                 4
(%i13) mod(a^2+a,63);
(%o13)                                20
(%i14) a:5;
(%o14)                                 5
(%i15) mod(a^2+a,63);
(%o15)                                30
(%i16) a:2;
(%o16)                                 2
(%i17) mod(a^2+a,63);
(%o17)                                 6
(%i18) a:6;
(%o18)                                 6
(%i19) mod(a^2+a,63);
(%o19)                                42
(%i20) a:7;
(%o20)                                 7
(%i21) mod(a^2+a,63);
(%o21)                                56
(%i22) a:8;
(%o22)                                 8
(%i23) mod(a^2+a,63);
(%o23)                                 9
(%i24) a:9;
(%o24)                                 9
(%i25) mod(a^2+a,63);
(%o25)                                27
(%i26) a:10;
(%o26)                                10
(%i27) mod(a^2+a,63);
(%o27)                                47
(%i28) a:11;
(%o28)                                11
(%i29) mod(a^2+a,63);
(%o29)                                 6
(%i30) a:12;
(%o30)                                12
(%i31) mod(a^2+a,63);
(%o31)                                30
(%i32) a:13;
(%o32)                                13
(%i33) mod(a^2+a,63);
(%o33)                                56
(%i34) a:14;
(%o34)                                14
(%i35) mod(a^2+a,63);
(%o35)                                21
(%i36) a:15;
(%o36)                                15
(%i37) mod(a^2+a,63);
(%o37)                                51
(%i38) a:16;
(%o38)                                16
(%i39) mod(a^2+a,63);
(%o39)                                20
(%i40) a:17;
(%o40)                                17
(%i41) mod(a^2+a,63);
(%o41)                                54
(%i42) a:18;
(%o42)                                18
(%i43) mod(a^2+a,63);
(%o43)                                27
(%i44) a:19;
(%o44)                                19
(%i45) mod(a^2+a,63);
(%o45)                                 2
(%i46) a:20;
(%o46)                                20
(%i47) mod(a^2+a,63);
(%o47)                                42
(%i48) a:21;
(%o48)                                21
(%i49) mod(a^2+a,63);
(%o49)                                21
(%i50) a:22;
(%o50)                                22
(%i51) mod(a^2+a,63);
(%o51)                                 2
(%i52) a:23;
(%o52)                                23
(%i53) mod(a^2+a,63);
(%o53)                                48
(%i54) a:24;
(%o54)                                24
(%i55) mod(a^2+a,63);
(%o55)                                33
(%i56) a:25;
(%o56)                                25
(%i57) mod(a^2+a,63);
(%o57)                                20
(%i58) a:26;
(%o58)                                26
(%i59) mod(a^2+a,63);
(%o59)                                 9
(%i60) a:27;
(%o60)                                27
(%i61) mod(a^2+a,63);
(%o61)                                 0
(%i62) a:28;
(%o62)                                28
(%i63) mod(a^2+a,63);
(%o63)                                56
(%i64) a:29;
(%o64)                                29
(%i65) mod(a^2+a,63);
(%o65)                                51
(%i66) a:30;
(%o66)                                30
(%i67) mod(a^2+a,63);
(%o67)                                48
(%i68) a:31;
(%o68)                                31
(%i69) mod(a^2+a,63);
(%o69)                                47
(%i70) a:32;
(%o70)                                32
(%i71) mod(a^2+a,63);
(%o71)                                48
mod(41,63)=41より、mod(x(x+1),63)=22でなければならない。しかし、mod(x(x+1),63)≠22であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 6日(水)19時10分48秒
返信・引用
  (%i1) solve(p^2+p+41=k*61,p);
                  sqrt(244 k - 163) + 1      sqrt(244 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(244*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        61
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,61);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,61);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,61);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,61);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,61);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,61);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,61);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,61);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,61);
(%o20)                                11
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,61);
(%o22)                                29
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,61);
(%o24)                                49
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,61);
(%o26)                                10
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,61);
(%o28)                                34
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,61);
(%o30)                                60
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,61);
(%o32)                                27
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,61);
(%o34)                                57
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,61);
(%o36)                                28
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,61);
(%o38)                                 1
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,61);
(%o40)                                37
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,61);
(%o42)                                14
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,61);
(%o44)                                54
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,61);
(%o46)                                35
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,61);
(%o48)                                18
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,61);
(%o50)                                 3
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,61);
(%o52)                                51
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,61);
(%o54)                                40
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,61);
(%o56)                                31
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,61);
(%o58)                                24
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,61);
(%o60)                                19
(%i61) a:29;
(%o61)                                29
(%i62) mod(a^2+a,61);
(%o62)                                16
(%i63) a:30;
(%o63)                                30
(%i64) mod(a^2+a,61);
(%o64)                                15
(%i65) a:31;
(%o65)                                31
(%i66) mod(a^2+a,61);
(%o66)                                16
mod(41,61)=41より、mod(x(x+1),61)=20でなければならない。mod(x(x+1),61)=20があり、kは整数になる場合がある。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 5日(火)19時35分27秒
返信・引用
  (%i1) solve(p^2+p+41=k*59,p);
                  sqrt(236 k - 163) + 1      sqrt(236 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(236*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        59
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,59);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,59);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,59);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,59);
(%o10)                                12
(%i12) a:4;
(%o12)                                 4
(%i13) mod(a^2+a,59);
(%o13)                                20
(%i14) a:5;
(%o14)                                 5
(%i15) mod(a^2+a,59);
(%o15)                                30
(%i16) a:6;
(%o16)                                 6
(%i17) mod(a^2+a,59);
(%o17)                                42
(%i18) a:7;
(%o18)                                 7
(%i19) mod(a^2+a,59);
(%o19)                                56
(%i20) a:8;
(%o20)                                 8
(%i21) mod(a^2+a,59);
(%o21)                                13
(%i22) a:9;
(%o22)                                 9
(%i23) mod(a^2+a,59);
(%o23)                                31
(%i24) a:10;
(%o24)                                10
(%i25) mod(a^2+a,59);
(%o25)                                51
(%i26) a:11;
(%o26)                                11
(%i27) mod(a^2+a,59);
(%o27)                                14
(%i28) a:12;
(%o28)                                12
(%i29) mod(a^2+a,59);
(%o29)                                38
(%i30) a:13;
(%o30)                                13
(%i31) mod(a^2+a,59);
(%o31)                                 5
(%i32) a:14;
(%o32)                                14
(%i33) mod(a^2+a,59);
(%o33)                                33
(%i34) a:15;
(%o34)                                15
(%i35) mod(a^2+a,59);
(%o35)                                 4
(%i36) a:16;
(%o36)                                16
(%i37) mod(a^2+a,59);
(%o37)                                36
(%i38) a:17;
(%o38)                                17
(%i39) mod(a^2+a,59);
(%o39)                                11
(%i40) a:18;
(%o40)                                18
(%i41) mod(a^2+a,59);
(%o41)                                47
(%i42) a:19;
(%o42)                                19
(%i43) mod(a^2+a,59);
(%o43)                                26
(%i44) a:20;
(%o44)                                20
(%i45) mod(a^2+a,59);
(%o45)                                 7
(%i46) a:21;
(%o46)                                21
(%i47) mod(a^2+a,59);
(%o47)                                49
(%i48) a:22;
(%o48)                                22
(%i49) mod(a^2+a,59);
(%o49)                                34
(%i50) a:23;
(%o50)                                23
(%i51) mod(a^2+a,59);
(%o51)                                21
(%i52) a:24;
(%o52)                                24
(%i53) mod(a^2+a,59);
(%o53)                                10
(%i54) a:25;
(%o54)                                25
(%i55) mod(a^2+a,59);
(%o55)                                 1
(%i56) a:26;
(%o56)                                26
(%i57) mod(a^2+a,59);
(%o57)                                53
(%i58) a:27;
(%o58)                                27
(%i59) mod(a^2+a,59);
(%o59)                                48
(%i60) a:28;
(%o60)                                28
(%i61) mod(a^2+a,59);
(%o61)                                45
(%i62) a:29;
(%o62)                                29
(%i63) mod(a^2+a,59);
(%o63)                                44
(%i64) a:30;
(%o64)                                30
(%i65) mod(a^2+a,59);
(%o65)                                45
mod(41,59)=41より、mod(x(x+1),59)=18でなければならない。しかしmod(x(x+1),59)≠18であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 4日(月)19時55分12秒
返信・引用
  (%i1) solve(p^2+p+41=k*57,p);
                  sqrt(228 k - 163) + 1      sqrt(228 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(228*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        57
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,57);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,57);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,57);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,57);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,57);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,57);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,57);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,57);
(%o18)                                56
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,57);
(%o20)                                15
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,57);
(%o22)                                33
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,57);
(%o24)                                53
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,57);
(%o26)                                18
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,57);
(%o28)                                42
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,57);
(%o30)                                11
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,57);
(%o32)                                39
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,57);
(%o34)                                12
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,57);
(%o36)                                44
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,57);
(%o38)                                21
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,57);
(%o40)                                 0
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,57);
(%o42)                                38
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,57);
(%o44)                                21
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,57);
(%o46)                                 6
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,57);
(%o48)                                50
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,57);
(%o50)                                39
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,57);
(%o52)                                30
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,57);
(%o54)                                23
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,57);
(%o56)                                18
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,57);
(%o58)                                15
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,57);
(%o60)                                14
(%i61) a:29;
(%o61)                                29
(%i62) mod(a^2+a,57);
(%o62)                                15
mod(41,57)=41より、mod(x(x+1),57)=16でなければならない。しかしmod(x(x+1),57)≠16であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 3日(日)19時07分4秒
返信・引用
  (%i1) solve(p^2+p+41=k*55,p);
                  sqrt(220 k - 163) + 1      sqrt(220 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(220*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        55
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,55);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,55);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,55);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,55);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,55);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,55);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,55);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,55);
(%o18)                                 1
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,55);
(%o20)                                17
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,55);
(%o22)                                35
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,55);
(%o24)                                 0
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,55);
(%o26)                                22
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,55);
(%o28)                                46
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,55);
(%o30)                                17
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,55);
(%o32)                                45
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,55);
(%o34)                                20
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,55);
(%o36)                                52
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,55);
(%o38)                                31
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,55);
(%o40)                                12
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,55);
(%o42)                                50
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,55);
(%o44)                                35
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,55);
(%o46)                                22
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,55);
(%o48)                                11
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,55);
(%o50)                                 2
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,55);
(%o52)                                50
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,55);
(%o54)                                45
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,55);
(%o56)                                42
(%i57) a:27;
(%o57)                                27
(%i58) mod(a^2+a,55);
(%o58)                                41
(%i59) a:28;
(%o59)                                28
(%i60) mod(a^2+a,55);
(%o60)                                42
mod(41,55)=41より、mod(x(x+1),55)=14でなければならない。しかしmod(x(x+1),55)≠14であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 2日(土)19時28分6秒
返信・引用
  (%i1) solve(p^2+p+41=k*53,p);
                  sqrt(212 k - 163) + 1      sqrt(212 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(212*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        53
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,53);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,53);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,53);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,53);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,53);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,53);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,53);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,53);
(%o18)                                 3
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,53);
(%o20)                                19
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,53);
(%o22)                                37
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,53);
(%o24)                                 4
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,53);
(%o26)                                26
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,53);
(%o28)                                50
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,53);
(%o30)                                23
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,53);
(%o32)                                51
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,53);
(%o34)                                28
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,53);
(%o36)                                 7
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,53);
(%o38)                                41
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,53);
(%o40)                                24
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,53);
(%o42)                                 9
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,53);
(%o44)                                49
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,53);
(%o46)                                38
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,53);
(%o48)                                29
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,53);
(%o50)                                22
mod(41,53)=41より、mod(x(x+1),53)=12でなければならない。mod(x(x+1),53)=12であり、kは整数になる場合がある。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 2日(土)14時58分16秒
返信・引用
  (%i1) solve(p^2+p+41=k*51,p);
                  sqrt(204 k - 163) + 1      sqrt(204 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(204*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        51
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,51);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,51);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,51);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,51);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,51);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,51);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,51);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,51);
(%o18)                                 5
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,51);
(%o20)                                21
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,51);
(%o22)                                39
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,51);
(%o24)                                 8
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,51);
(%o26)                                30
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,51);
(%o28)                                 3
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,51);
(%o30)                                29
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,51);
(%o32)                                 6
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,51);
(%o34)                                36
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,51);
(%o36)                                17
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,51);
(%o38)                                 0
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,51);
(%o40)                                36
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,51);
(%o42)                                23
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,51);
(%o44)                                12
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,51);
(%o46)                                 3
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,51);
(%o48)                                47
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,51);
(%o50)                                42
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,51);
(%o52)                                39
(%i53) a:25;
(%o53)                                25
(%i54) mod(a^2+a,51);
(%o54)                                38
(%i55) a:26;
(%o55)                                26
(%i56) mod(a^2+a,51);
(%o56)                                39
(%i57)
mod(41,51)=41より、mod(x(x+1),51)=10でなければならない。しかし、mod(x(x+1),51)≠10であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 6月 1日(金)16時27分55秒
返信・引用
  (%i1) solve(p^2+p+41=k*49,p);
                  sqrt(196 k - 163) + 1      sqrt(196 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(196*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        49
において、
(%i8) a:0;
(%o8)                                  0
(%i9) mod(a^2+a,49);
(%o9)                                  0
(%i10) a:1;
(%o10)                                 1
(%i11) mod(a^2+a,49);
(%o11)                                 2
(%i12) a:2;
(%o12)                                 2
(%i13) mod(a^2+a,49);
(%o13)                                 6
(%i14) a:3;
(%o14)                                 3
(%i15) mod(a^2+a,49);
(%o15)                                12
(%i16) a:4;
(%o16)                                 4
(%i17) mod(a^2+a,49);
(%o17)                                20
(%i18) a:5;
(%o18)                                 5
(%i19) mod(a^2+a,49);
(%o19)                                30
(%i20) a:6;
(%o20)                                 6
(%i21) mod(a^2+a,49);
(%o21)                                42
(%i22) a:7;
(%o22)                                 7
(%i23) mod(a^2+a,49);
(%o23)                                 7
(%i24) a:8;
(%o24)                                 8
(%i25) mod(a^2+a,49);
(%o25)                                23
(%i26) a:9;
(%o26)                                 9
(%i27) mod(a^2+a,49);
(%o27)                                41
(%i30) a:10;
(%o30)                                10
(%i31) mod(a^2+a,49);
(%o31)                                12
(%i28) a:11;
(%o28)                                11
(%i29) mod(a^2+a,49);
(%o29)                                34
(%i32) a:12;
(%o32)                                12
(%i33) mod(a^2+a,49);
(%o33)                                 9
(%i34) a:13;
(%o34)                                13
(%i35) mod(a^2+a,49);
(%o35)                                35
(%i36) a:14;
(%o36)                                14
(%i37) mod(a^2+a,49);
(%o37)                                14
(%i38) a:15;
(%o38)                                15
(%i39) mod(a^2+a,49);
(%o39)                                44
(%i40) a:16;
(%o40)                                16
(%i41) mod(a^2+a,49);
(%o41)                                27
(%i42) a:17;
(%o42)                                17
(%i43) mod(a^2+a,49);
(%o43)                                12
(%i44) a:18;
(%o44)                                18
(%i45) mod(a^2+a,49);
(%o45)                                48
(%i46) a:19;
(%o46)                                19
(%i47) mod(a^2+a,49);
(%o47)                                37
(%i48) a:20;
(%o48)                                20
(%i49) mod(a^2+a,49);
(%o49)                                28
(%i50) a:21;
(%o50)                                21
(%i51) mod(a^2+a,49);
(%o51)                                21
(%i52) a:22;
(%o52)                                22
(%i53) mod(a^2+a,49);
(%o53)                                16
(%i54) a:23;
(%o54)                                23
(%i55) mod(a^2+a,49);
(%o55)                                13
(%i56) a:24;
(%o56)                                24
(%i57) mod(a^2+a,49);
(%o57)                                12
(%i58) a:25;
(%o58)                                25
(%i59) mod(a^2+a,49);
(%o59)                                13

mod(41,49)=41より、mod(x(x+1),49)=8でなければならない。しかし、mod(x(x+1),49)≠8であり、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月28日(月)19時37分3秒
返信・引用
  (%i1) solve(p^2+p+41=k*47,p);
                  sqrt(188 k - 163) + 1      sqrt(188 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(188*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        47
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,47);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,47);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,47);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,47);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,47);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,47);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,47);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,47);
(%o18)                                 9
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,47);
(%o20)                                25
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,47);
(%o22)                                43
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,47);
(%o24)                                16
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,47);
(%o26)                                38
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,47);
(%o28)                                15
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,47);
(%o30)                                41
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,47);
(%o32)                                22
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,47);
(%o34)                                 5
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,47);
(%o36)                                37
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,47);
(%o38)                                24
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,47);
(%o40)                                13
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,47);
(%o42)                                 4
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,47);
(%o44)                                44
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,47);
(%o46)                                39
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,47);
(%o48)                                36
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,47);
(%o50)                                35
(%i51) a:24;
(%o51)                                24
(%i52) mod(a^2+a,47);
(%o52)                                36

mod(41,47)=41より、mod(x(x+1),47)=6でなければならない。mod(x(x+1),47)=6があり、kは整数になる場合がある。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月27日(日)11時36分15秒
返信・引用
  (%i1) solve(p^2+p+41=k*45,p);
                  sqrt(180 k - 163) + 1      sqrt(180 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(180*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        45
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,45);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,45);
(%o6)                                  2
(%i7) a:2;
(%o7)                                  2
(%i8) mod(a^2+a,45);
(%o8)                                  6
(%i9) a:3;
(%o9)                                  3
(%i10) mod(a^2+a,45);
(%o10)                                12
(%i11) a:4;
(%o11)                                 4
(%i12) mod(a^2+a,45);
(%o12)                                20
(%i13) a:5;
(%o13)                                 5
(%i14) mod(a^2+a,45);
(%o14)                                30
(%i15) a:6;
(%o15)                                 6
(%i16) mod(a^2+a,45);
(%o16)                                42
(%i17) a:7;
(%o17)                                 7
(%i18) mod(a^2+a,45);
(%o18)                                11
(%i19) a:8;
(%o19)                                 8
(%i20) mod(a^2+a,45);
(%o20)                                27
(%i21) a:9;
(%o21)                                 9
(%i22) mod(a^2+a,45);
(%o22)                                 0
(%i23) a:10;
(%o23)                                10
(%i24) mod(a^2+a,45);
(%o24)                                20
(%i25) a:11;
(%o25)                                11
(%i26) mod(a^2+a,45);
(%o26)                                42
(%i27) a:12;
(%o27)                                12
(%i28) mod(a^2+a,45);
(%o28)                                21
(%i29) a:13;
(%o29)                                13
(%i30) mod(a^2+a,45);
(%o30)                                 2
(%i31) a:14;
(%o31)                                14
(%i32) mod(a^2+a,45);
(%o32)                                30
(%i33) a:15;
(%o33)                                15
(%i34) mod(a^2+a,45);
(%o34)                                15
(%i35) a:16;
(%o35)                                16
(%i36) mod(a^2+a,45);
(%o36)                                 2
(%i37) a:17;
(%o37)                                17
(%i38) mod(a^2+a,45);
(%o38)                                36
(%i39) a:18;
(%o39)                                18
(%i40) mod(a^2+a,45);
(%o40)                                27
(%i41) a:19;
(%o41)                                19
(%i42) mod(a^2+a,45);
(%o42)                                20
(%i43) a:20;
(%o43)                                20
(%i44) mod(a^2+a,45);
(%o44)                                15
(%i45) a:21;
(%o45)                                21
(%i46) mod(a^2+a,45);
(%o46)                                12
(%i47) a:22;
(%o47)                                22
(%i48) mod(a^2+a,45);
(%o48)                                11
(%i49) a:23;
(%o49)                                23
(%i50) mod(a^2+a,45);
(%o50)                                12
mod(41,45)=41より、mod(x(x+1),45)=4でなければならない。しかしmod(x(x+1),45)≠4より、kは整数にならない。
 

Re: 有理数と無理数

 投稿者:うんざりはちべえ  投稿日:2018年 5月27日(日)11時26分30秒
返信・引用
  > もちろん、3.1415926・・・・=xとすると言えば、代数である。演算はできる。

しかし、有理数は有理数で閉じているから、無理数にはならない。
 

Re: 有理数と無理数

 投稿者:うんざりはちべえ  投稿日:2018年 5月26日(土)19時44分18秒
返信・引用
  > 有理数は4則において閉じているので、
> ライプニッツの公式https://ja.wikipedia.org/wiki/%E3%83%A9%E3%82%A4%E3%83%97%E3%83%8B%E3%83%83%E3%83%84%E3%81%AE%E5%85%AC%E5%BC%8F
> は、誤りである。なぜなら、左辺は有理数右辺は無理数であるから等号で結ぶことはできない。

たとえば、πは、3.1415926・・・・を表すラベルとしよう。
すると、ζ(2)の答えを表すπ^2/6というラベルであるとできる。

ラベルなんだというとラベル同士の演算はできるのであろうか?
ζ(2)/π=π/6
という具合に・・・

もっといえば、無限大のラベルは∞である。ラベル同士の演算はできない。

もちろん、3.1415926・・・・=xとすると言えば、代数である。演算はできる。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月26日(土)19時33分59秒
返信・引用
  (%i1) solve(p^2+p+41=k*43,p);
                  sqrt(172 k - 163) + 1      sqrt(172 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(172*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        43
において、
(%i3) a:0;
(%o3)                                  0
(%i4) mod(a^2+a,43);
(%o4)                                  0
(%i5) a:1;
(%o5)                                  1
(%i6) mod(a^2+a,43);
(%o6)                                  2
mod(41,43)=41より、mod(x(x+1),43)=2でなければならない。mod(x(x+1),43)=0,2,・・・より、kは整数になる場合がある。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 8日(火)19時27分50秒
返信・引用 編集済
  x=41n+aとすると、
(%i1) x:41*n+a;
(%o1)                              41 n + a
(%i2) expand(x*(x+1));
                             2                    2
(%o2)                  1681 n  + 82 a n + 41 n + a  + a
41の倍数でないのはa^2+aである。
したがって、
mod(x*(x+1),41)=mod(a^2+a,41)=mod(a(a+1),41)
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 8日(火)19時16分55秒
返信・引用
  どれも、41なら、41n+20でそれまでとは、鏡でうつしたように、mod(x*(x+1),41)は逆戻りする。  

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 7日(月)14時33分17秒
返信・引用
  (%i1) solve(p^2+p+41=k*41,p);
                  sqrt(164 k - 163) + 1      sqrt(164 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(164*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        41
において、
(%i3) x:41*n;
(%o3)                                41 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                           1681 n  + 41 n
(%i5) x:41*n+1;
(%o5)                              41 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                         1681 n  + 123 n + 2
(%i7) x:41*n+2;
(%o7)                              41 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                         1681 n  + 205 n + 6
(%i9) x:41*n+3;
(%o9)                              41 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                       1681 n  + 287 n + 12
(%i11) x:41*n+4;
(%o11)                             41 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                       1681 n  + 369 n + 20
(%i13) x:41*n+5;
(%o13)                             41 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                       1681 n  + 451 n + 30
(%i15) x:41*n+6;
(%o15)                             41 n + 6
(%i16) expand(x*(x+1));
                                   2
(%o16)                       1681 n  + 533 n + 42
(%i17) mod(42,41);
(%o17)                                 1
(%i18) x:41*n+7;
(%o18)                             41 n + 7
(%i19) expand(x*(x+1));
                                   2
(%o19)                       1681 n  + 615 n + 56
(%i20) mod(56,41);
(%o20)                                15
(%i21) x:41*n+8;
(%o21)                             41 n + 8
(%i22) expand(x*(x+1));
                                   2
(%o22)                       1681 n  + 697 n + 72
(%i23) mod(72,41);
(%o23)                                31
(%i24) x:41*n+9;
(%o24)                             41 n + 9
(%i25) expand(x*(x+1));
                                   2
(%o25)                       1681 n  + 779 n + 90
(%i26) mod(90,41);
(%o26)                                 8
(%i27) x:41*n+10;
(%o27)                             41 n + 10
(%i28) expand(x*(x+1));
                                   2
(%o28)                       1681 n  + 861 n + 110
(%i29) mod(110,41);
(%o29)                                28
(%i30) x:41*n+11;
(%o30)                             41 n + 11
(%i31) expand(x*(x+1));
                                   2
(%o31)                       1681 n  + 943 n + 132
(%i32) mod(132,41);
(%o32)                                 9
(%i33) x:41*n+12;
(%o33)                             41 n + 12
(%i34) expand(x*(x+1));
                                  2
(%o34)                      1681 n  + 1025 n + 156
(%i35) mod(156,41);
(%o35)                                33
(%i36) x:41*n+13;
(%o36)                             41 n + 13
(%i37) expand(x*(x+1));
                                  2
(%o37)                      1681 n  + 1107 n + 182
(%i38) mod(182,41);
(%o38)                                18
(%i39) x:41*n+14;
(%o39)                             41 n + 14
(%i40) expand(x*(x+1));
                                  2
(%o40)                      1681 n  + 1189 n + 210
(%i41) mod(210,41);
(%o41)                                 5
(%i42) x:41*n+15;
(%o42)                             41 n + 15
(%i43) expand(x*(x+1));
                                  2
(%o43)                      1681 n  + 1271 n + 240
(%i44) mod(240,41);
(%o44)                                35
(%i45) x:41*n+16;
(%o45)                             41 n + 16
(%i46) expand(x*(x+1));
                                  2
(%o46)                      1681 n  + 1353 n + 272
(%i47) mod(272,41);
(%o47)                                26
(%i48) x:41*n+17;
(%o48)                             41 n + 17
(%i49) expand(x*(x+1));
                                  2
(%o49)                      1681 n  + 1435 n + 306
(%i50) mod(306,41);
(%o50)                                19
(%i51) x:41*n+18;
(%o51)                             41 n + 18
(%i52) expand(x*(x+1));
                                  2
(%o52)                      1681 n  + 1517 n + 342
(%i53) mod(342,41);
(%o53)                                14
(%i54) x:41*n+19;
(%o54)                             41 n + 19
(%i55) expand(x*(x+1));
                                  2
(%o55)                      1681 n  + 1599 n + 380
(%i56) mod(380,41);
(%o56)                                11
(%i57) x:41*n+20;
(%o57)                             41 n + 20
(%i58) expand(x*(x+1));
                                  2
(%o58)                      1681 n  + 1681 n + 420
(%i59) mod(420,41);
(%o59)                                10
(%i60) x:41*n+21;
(%o60)                             41 n + 21
(%i61) expand(x*(x+1));
                                  2
(%o61)                      1681 n  + 1763 n + 462
(%i62) mod(462,41);
(%o62)                                11
(%i63) x:41*n+22;
(%o63)                             41 n + 22
(%i64) expand(x*(x+1));
                                  2
(%o64)                      1681 n  + 1845 n + 506
(%i65) mod(506,41);
(%o65)                                14
(%i66) x:41*n+23;
(%o66)                             41 n + 23
(%i67) expand(x*(x+1));
                                  2
(%o67)                      1681 n  + 1927 n + 552
(%i68) mod(552,41);
(%o68)                                19
(%i69) x:41*n+24;
(%o69)                             41 n + 24
(%i70) expand(x*(x+1));
                                  2
(%o70)                      1681 n  + 2009 n + 600
(%i71) mod(600,41);
(%o71)                                26
(%i72) x:41*n+25;
(%o72)                             41 n + 25
(%i73) expand(x*(x+1));
                                  2
(%o73)                      1681 n  + 2091 n + 650
(%i74) mod(650,41);
(%o74)                                35
(%i75) x:41*n+26;
(%o75)                             41 n + 26
(%i76) expand(x*(x+1));
                                  2
(%o76)                      1681 n  + 2173 n + 702
(%i77) mod(702,41);
(%o77)                                 5
(%i78) x:41*n+27;
(%o78)                             41 n + 27
(%i79) expand(x*(x+1));
                                  2
(%o79)                      1681 n  + 2255 n + 756
(%i80) mod(756,41);
(%o80)                                18
(%i81) x:41*n+28;
(%o81)                             41 n + 28
(%i82) expand(x*(x+1));
                                  2
(%o82)                      1681 n  + 2337 n + 812
(%i83) mod(812,41);
(%o83)                                33
(%i84) x:41*n+29;
(%o84)                             41 n + 29
(%i85) expand(x*(x+1));
                                  2
(%o85)                      1681 n  + 2419 n + 870
(%i86) mod(870,41);
(%o86)                                 9
(%i87) x:41*n+30;
(%o87)                             41 n + 30
(%i88) expand(x*(x+1));
                                  2
(%o88)                      1681 n  + 2501 n + 930
(%i89) mod(930,41);
(%o89)                                28
(%i90) x:41*n+31;
(%o90)                             41 n + 31
(%i91) expand(x*(x+1));
                                  2
(%o91)                      1681 n  + 2583 n + 992
(%i92) mod(992,41);
(%o92)                                 8
(%i93) x:41*n+32;
(%o93)                             41 n + 32
(%i94) expand(x*(x+1));
                                  2
(%o94)                      1681 n  + 2665 n + 1056
(%i95) mod(1056,41);
(%o95)                                31
(%i96) x:41*n+33;
(%o96)                             41 n + 33
(%i97) expand(x*(x+1));
                                  2
(%o97)                      1681 n  + 2747 n + 1122
(%i98) mod(1122,41);
(%o98)                                15
(%i99) x:41*n+34;
(%o99)                             41 n + 34
(%i100) expand(x*(x+1));
                                  2
(%o100)                     1681 n  + 2829 n + 1190
(%i101) mod(1190,41);
(%o101)                                1
(%i102) x:41*n+35;
(%o102)                            41 n + 35
(%i103) expand(x*(x+1));
                                  2
(%o103)                     1681 n  + 2911 n + 1260
(%i104) mod(1260,41);
(%o104)                               30
(%i105) x:41*n+36;
(%o105)                            41 n + 36
(%i106) expand(x*(x+1));
                                  2
(%o106)                     1681 n  + 2993 n + 1332
(%i107) mod(1332,41);
(%o107)                               20
(%i108) x:41*n+37;
(%o108)                            41 n + 37
(%i109) expand(x*(x+1));
                                  2
(%o109)                     1681 n  + 3075 n + 1406
(%i110) mod(1406,41);
(%o110)                               12
(%i111) x:41*n+38;
(%o111)                            41 n + 38
(%i112) expand(x*(x+1));
                                  2
(%o112)                     1681 n  + 3157 n + 1482
(%i113) mod(1482,41);
(%o113)                                6
(%i114) x:41*n+39;
(%o114)                            41 n + 39
(%i115) expand(x*(x+1));
                                  2
(%o115)                     1681 n  + 3239 n + 1560
(%i116) mod(1560,41);
(%o116)                                2
(%i117) x:41*n+40;
(%o117)                            41 n + 40
(%i118) expand(x*(x+1));
                                  2
(%o118)                     1681 n  + 3321 n + 1640
(%i119) mod(1640,41);
(%o119)                                0
mod(41,41)=0より、mod(x(x+1),39)=0でなければならない。mod(x(x+1),41)=0,1,2,5,8,9,10,11,12,14,15,18,19,20,26,28,30,31,33,35から0になる場合があるのでkは整数になる場合がある。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 7日(月)14時32分35秒
返信・引用
  (%i1) solve(p^2+p+41=k*39,p);
                  sqrt(156 k - 163) + 1      sqrt(156 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(156*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        39
において、
(%i3) x:39*n;
(%o3)                                39 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                           1521 n  + 39 n
(%i5) x:39*n+1;
(%o5)                              39 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                         1521 n  + 117 n + 2
(%i7) x:39*n+2;
(%o7)                              39 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                         1521 n  + 195 n + 6
(%i9) x:39*n+3;
(%o9)                              39 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                       1521 n  + 273 n + 12
(%i11) x:39*n+4;
(%o11)                             39 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                       1521 n  + 351 n + 20
(%i13) x:39*n+5;
(%o13)                             39 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                       1521 n  + 429 n + 30
(%i15) x:39*n+6;
(%o15)                             39 n + 6
(%i16) expand(x*(x+1));
                                   2
(%o16)                       1521 n  + 507 n + 42
(%i17) mod(42,39);
(%o17)                                 3
(%i18) x:39*n+7;
(%o18)                             39 n + 7
(%i19) expand(x*(x+1));
                                   2
(%o19)                       1521 n  + 585 n + 56
(%i20) mod(56,39);
(%o20)                                17
(%i21) x:39*n+8;
(%o21)                             39 n + 8
(%i22) expand(x*(x+1));
                                   2
(%o22)                       1521 n  + 663 n + 72
(%i23) mod(72,39);
(%o23)                                33
(%i24) x:39*n+9;
(%o24)                             39 n + 9
(%i25) expand(x*(x+1));
                                   2
(%o25)                       1521 n  + 741 n + 90
(%i26) mod(90,39);
(%o26)                                12
(%i27) x:39*n+10;
(%o27)                             39 n + 10
(%i28) expand(x*(x+1));
                                   2
(%o28)                       1521 n  + 819 n + 110
(%i29) mod(110,39);
(%o29)                                32
(%i30) x:39*n+11;
(%o30)                             39 n + 11
(%i31) expand(x*(x+1));
                                   2
(%o31)                       1521 n  + 897 n + 132
(%i32) mod(132,39);
(%o32)                                15
(%i33) x:39*n+12;
(%o33)                             39 n + 12
(%i34) expand(x*(x+1));
                                   2
(%o34)                       1521 n  + 975 n + 156
(%i35) mod(156,39);
(%o35)                                 0
(%i36) x:39*n+13;
(%o36)                             39 n + 13
(%i37) expand(x*(x+1));
                                  2
(%o37)                      1521 n  + 1053 n + 182
(%i38) mod(182,39);
(%o38)                                26
(%i39) x:39*n+14;
(%o39)                             39 n + 14
(%i40) expand(x*(x+1));
                                  2
(%o40)                      1521 n  + 1131 n + 210
(%i41) mod(210,39);
(%o41)                                15
(%i42) x:39*n+15;
(%o42)                             39 n + 15
(%i43) expand(x*(x+1));
                                  2
(%o43)                      1521 n  + 1209 n + 240
(%i44) mod(240,39);
(%o44)                                 6
(%i45) x:39*n+16;
(%o45)                             39 n + 16
(%i46) expand(x*(x+1));
                                  2
(%o46)                      1521 n  + 1287 n + 272
(%i47) mod(272,39);
(%o47)                                38
(%i48) x:39*n+17;
(%o48)                             39 n + 17
(%i49) expand(x*(x+1));
                                  2
(%o49)                      1521 n  + 1365 n + 306
(%i50) mod(306,39);
(%o50)                                33
(%i51) x:39*n+18;
(%o51)                             39 n + 18
(%i52) expand(x*(x+1));
                                  2
(%o52)                      1521 n  + 1443 n + 342
(%i53) mod(342,39);
(%o53)                                30
(%i54) x:39*n+19;
(%o54)                             39 n + 19
(%i55) expand(x*(x+1));
                                  2
(%o55)                      1521 n  + 1521 n + 380
(%i56) mod(380,39);
(%o56)                                29
(%i57) x:39*n+20;
(%o57)                             39 n + 20
(%i58) expand(x*(x+1));
                                  2
(%o58)                      1521 n  + 1599 n + 420
(%i59) mod(420,39);
(%o59)                                30
(%i60) x:39*n+21;
(%o60)                             39 n + 21
(%i61) expand(x*(x+1));
                                  2
(%o61)                      1521 n  + 1677 n + 462
(%i62) mod(462,39);
(%o62)                                33
(%i63) x:39*n+22;
(%o63)                             39 n + 22
(%i64) expand(x*(x+1));
                                  2
(%o64)                      1521 n  + 1755 n + 506
(%i65) mod(506,39);
(%o65)                                38
(%i66) x:39*n+23;
(%o66)                             39 n + 23
(%i67) expand(x*(x+1));
                                  2
(%o67)                      1521 n  + 1833 n + 552
(%i68) mod(552,39);
(%o68)                                 6
(%i69) x:39*n+24;
(%o69)                             39 n + 24
(%i70) expand(x*(x+1));
                                  2
(%o70)                      1521 n  + 1911 n + 600
(%i71) mod(600,39);
(%o71)                                15
(%i72) x:39*n+25;
(%o72)                             39 n + 25
(%i73) expand(x*(x+1));
                                  2
(%o73)                      1521 n  + 1989 n + 650
(%i74) mod(650,39);
(%o74)                                26
(%i75) x:39*n+26;
(%o75)                             39 n + 26
(%i76) expand(x*(x+1));
                                  2
(%o76)                      1521 n  + 2067 n + 702
(%i77) mod(702,39);
(%o77)                                 0
(%i78) x:39*n+27;
(%o78)                             39 n + 27
(%i79) expand(x*(x+1));
                                  2
(%o79)                      1521 n  + 2145 n + 756
(%i80) mod(756,39);
(%o80)                                15
(%i81) x:39*n+28;
(%o81)                             39 n + 28
(%i82) expand(x*(x+1));
                                  2
(%o82)                      1521 n  + 2223 n + 812
(%i83) mod(812,39);
(%o83)                                32
(%i84) x:39*n+29;
(%o84)                             39 n + 29
(%i85) expand(x*(x+1));
                                  2
(%o85)                      1521 n  + 2301 n + 870
(%i86) mod(870,39);
(%o86)                                12
(%i87) x:39*n+30;
(%o87)                             39 n + 30
(%i88) expand(x*(x+1));
                                  2
(%o88)                      1521 n  + 2379 n + 930
(%i89) mod(930,39);
(%o89)                                33
(%i90) x:39*n+31;
(%o90)                             39 n + 31
(%i91) expand(x*(x+1));
                                  2
(%o91)                      1521 n  + 2457 n + 992
(%i92) mod(992,39);
(%o92)                                17
(%i93) x:39*n+32;
(%o93)                             39 n + 32
(%i94) expand(x*(x+1));
                                  2
(%o94)                      1521 n  + 2535 n + 1056
(%i95) mod(1056,39);
(%o95)                                 3
(%i96) x:39*n+33;
(%o96)                             39 n + 33
(%i97) expand(x*(x+1));
                                  2
(%o97)                      1521 n  + 2613 n + 1122
(%i98) mod(1122,39);
(%o98)                                30
(%i99) x:39*n+34;
(%o99)                             39 n + 34
(%i100) expand(x*(x+1));
                                  2
(%o100)                     1521 n  + 2691 n + 1190
(%i101) mod(1190,39);
(%o101)                               20
(%i102) x:39*n+35;
(%o102)                            39 n + 35
(%i103) expand(x*(x+1));
                                  2
(%o103)                     1521 n  + 2769 n + 1260
(%i104) mod(1260,39);
(%o104)                               12
(%i105) x:39*n+36;
(%o105)                            39 n + 36
(%i106) expand(x*(x+1));
                                  2
(%o106)                     1521 n  + 2847 n + 1332
(%i107) mod(1332,39);
(%o107)                                6
(%i108) x:39*n+37;
(%o108)                            39 n + 37
(%i109) expand(x*(x+1));
                                  2
(%o109)                     1521 n  + 2925 n + 1406
(%i110) mod(1406,39);
(%o110)                                2
(%i111) x:39*n+38;
(%o111)                            39 n + 38
(%i112) expand(x*(x+1));
                                  2
(%o112)                     1521 n  + 3003 n + 1482
(%i113) mod(1482,39);
(%o113)                                0
(%i115) mod(41,39);
(%o115)                                2

mod(41,39)=2より、mod(x(x+1),39)=37でなければならない。しかし、mod(x(x+1),39)=0,2,3,6,10,11,12,15,17,20,26,29,30,32,33,38より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 6日(日)20時26分4秒
返信・引用
  (%i1) solve(p^2+p+41=k*37,p);
                  sqrt(148 k - 163) + 1      sqrt(148 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(148*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        37
において、
(%i3) x:37*n;
(%o3)                                37 n
(%i5) expand(x*(x+1));
                                      2
(%o5)                           1369 n  + 37 n
(%i6) x:37*n+1;
(%o6)                              37 n + 1
(%i7) expand(x*(x+1));
                                    2
(%o7)                         1369 n  + 111 n + 2
(%i8) x:37*n+2;
(%o8)                              37 n + 2
(%i9) expand(x*(x+1));
                                    2
(%o9)                         1369 n  + 185 n + 6
(%i10) x:37*n+3;
(%o10)                             37 n + 3
(%i11) expand(x*(x+1));
                                   2
(%o11)                       1369 n  + 259 n + 12
(%i12) x:37*n+4;
(%o12)                             37 n + 4
(%i13) expand(x*(x+1));
                                   2
(%o13)                       1369 n  + 333 n + 20
(%i14) x:37*n+5;
(%o14)                             37 n + 5
(%i15) expand(x*(x+1));
                                   2
(%o15)                       1369 n  + 407 n + 30
(%i16) x:37*n+6;
(%o16)                             37 n + 6
(%i17) expand(x*(x+1));
                                   2
(%o17)                       1369 n  + 481 n + 42
(%i18) mod(42,37);
(%o18)                                 5
(%i19) x:37*n+7;
(%o19)                             37 n + 7
(%i20) expand(x*(x+1));
                                   2
(%o20)                       1369 n  + 555 n + 56
(%i21) mod(56,37);
(%o21)                                19
(%i22) x:37*n+8;
(%o22)                             37 n + 8
(%i24) expand(x*(x+1));
                                   2
(%o24)                       1369 n  + 629 n + 72
(%i25) mod(72,37);
(%o25)                                35
(%i26) x:37*n+9;
(%o26)                             37 n + 9
(%i27) expand(x*(x+1));
                                   2
(%o27)                       1369 n  + 703 n + 90
(%i28) mod(90,37);
(%o28)                                16
(%i29) x:37*n+10;
(%o29)                             37 n + 10
(%i30) expand(x*(x+1));
                                   2
(%o30)                       1369 n  + 777 n + 110
(%i31) mod(110,37);
(%o31)                                36
(%i32) x:37*n+11;
(%o32)                             37 n + 11
(%i33) expand(x*(x+1));
                                   2
(%o33)                       1369 n  + 851 n + 132
(%i34) mod(132,37);
(%o34)                                21
(%i35) x:37*n+12;
(%o35)                             37 n + 12
(%i36) expand(x*(x+1));
                                   2
(%o36)                       1369 n  + 925 n + 156
(%i37) mod(156,37);
(%o37)                                 8
(%i38) x:37*n+13;
(%o38)                             37 n + 13
(%i39) expand(x*(x+1));
                                   2
(%o39)                       1369 n  + 999 n + 182
(%i40) mod(182,37);
(%o40)                                34
(%i41) x:37*n+14;
(%o41)                             37 n + 14
(%i42) expand(x*(x+1));
                                  2
(%o42)                      1369 n  + 1073 n + 210
(%i43) mod(210,37);
(%o43)                                25
(%i44) x:37*n+15;
(%o44)                             37 n + 15
(%i45) expand(x*(x+1));
                                  2
(%o45)                      1369 n  + 1147 n + 240
(%i46) mod(240,37);
(%o46)                                18
(%i47) x:37*n+16;
(%o47)                             37 n + 16
(%i48) expand(x*(x+1));
                                  2
(%o48)                      1369 n  + 1221 n + 272
(%i49) mod(272,37);
(%o49)                                13
(%i50) x:37*n+17;
(%o50)                             37 n + 17
(%i51) expand(x*(x+1));
                                  2
(%o51)                      1369 n  + 1295 n + 306
(%i52) mod(306,37);
(%o52)                                10
(%i53) x:37*n+18;
(%o53)                             37 n + 18
(%i54) expand(x*(x+1));
                                  2
(%o54)                      1369 n  + 1369 n + 342
(%i55) mod(342,37);
(%o55)                                 9
(%i56) x:37*n+19;
(%o56)                             37 n + 19
(%i57) expand(x*(x+1));
                                  2
(%o57)                      1369 n  + 1443 n + 380
(%i58) mod(380,37);
(%o58)                                10
(%i59) x:37*n+20;
(%o59)                             37 n + 20
(%i60) expand(x*(x+1));
                                  2
(%o60)                      1369 n  + 1517 n + 420
(%i61) mod(420,37);
(%o61)                                13
(%i62) x:37*n+21;
(%o62)                             37 n + 21
(%i63) expand(x*(x+1));
                                  2
(%o63)                      1369 n  + 1591 n + 462
(%i64) mod(462,37);
(%o64)                                18
(%i65) x:37*n+22;
(%o65)                             37 n + 22
(%i66) expand(x*(x+1));
                                  2
(%o66)                      1369 n  + 1665 n + 506
(%i67) mod(506,37);
(%o67)                                25
(%i68) x:37*n+23;
(%o68)                             37 n + 23
(%i69) expand(x*(x+1));
                                  2
(%o69)                      1369 n  + 1739 n + 552
(%i70) mod(552,37);
(%o70)                                34
(%i71) x:37*n+24;
(%o71)                             37 n + 24
(%i72) expand(x*(x+1));
                                  2
(%o72)                      1369 n  + 1813 n + 600
(%i73) mod(600,37);
(%o73)                                 8
(%i74) x:37*n+25;
(%o74)                             37 n + 25
(%i75) expand(x*(x+1));
                                  2
(%o75)                      1369 n  + 1887 n + 650
(%i76) mod(650,37);
(%o76)                                21
(%i77) x:37*n+26;
(%o77)                             37 n + 26
(%i78) expand(x*(x+1));
                                  2
(%o78)                      1369 n  + 1961 n + 702
(%i79) mod(702,37);
(%o79)                                36
(%i80) x:37*n+27;
(%o80)                             37 n + 27
(%i81) expand(x*(x+1));
                                  2
(%o81)                      1369 n  + 2035 n + 756
(%i82) mod(756,37);
(%o82)                                16
(%i83) x:37*n+28;
(%o83)                             37 n + 28
(%i84) expand(x*(x+1));
                                  2
(%o84)                      1369 n  + 2109 n + 812
(%i85) mod(812,37);
(%o85)                                35
(%i86) x:37*n+29;
(%o86)                             37 n + 29
(%i87) expand(x*(x+1));
                                  2
(%o87)                      1369 n  + 2183 n + 870
(%i88) mod(870,37);
(%o88)                                19
(%i89) x:37*n+30;
(%o89)                             37 n + 30
(%i90) expand(x*(x+1));
                                  2
(%o90)                      1369 n  + 2257 n + 930
(%i91) mod(930,37);
(%o91)                                 5
(%i92) x:37*n+31;
(%o92)                             37 n + 31
(%i93) expand(x*(x+1));
                                  2
(%o93)                      1369 n  + 2331 n + 992
(%i94) mod(992,37);
(%o94)                                30
(%i95) x:37*n+32;
(%o95)                             37 n + 32
(%i96) expand(x*(x+1));
                                  2
(%o96)                      1369 n  + 2405 n + 1056
(%i97) mod(1056,37);
(%o97)                                20
(%i98) x:37*n+33;
(%o98)                             37 n + 33
(%i99) expand(x*(x+1));
                                  2
(%o99)                      1369 n  + 2479 n + 1122
(%i100) mod(1122,37);
(%o100)                               12
(%i101) x:37*n+34;
(%o101)                            37 n + 34
(%i102) expand(x*(x+1));
                                  2
(%o102)                     1369 n  + 2553 n + 1190
(%i103) mod(1190,37);
(%o103)                                6
(%i104) x:37*n+35;
(%o104)                            37 n + 35
(%i105) expand(x*(x+1));
                                  2
(%o105)                     1369 n  + 2627 n + 1260
(%i106) mod(1260,37);
(%o106)                                2
(%i107) x:37*n+36;
(%o107)                            37 n + 36
(%i108) expand(x*(x+1));
                                  2
(%o108)                     1369 n  + 2701 n + 1332
(%i109) mod(1332,37);
(%o109)                                0
mod(41,37)=4より、mod(x(x+1),37)=33でなければならない。しかし、mod(x(x+1),37)=0,2,5,6,7,8,10,11,12,13,16,18,19,20,21,25,30,34,35,36より、kは整数にならない。
 

会計士の数学の応用

 投稿者:うんざりはちべえ  投稿日:2018年 5月 6日(日)19時30分9秒
返信・引用 編集済
  http://y-daisan.private.coocan.jp/html/kaikeisi.pdf
のf(x^2)が 初等整数論 遠山啓著 日本評論社1997年6月25日第1版第17刷発行の九去法のところで、113ページにf(x^2)≠5を証明せよとあるので、昔からわかっていたことですね。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 6日(日)19時18分11秒
返信・引用
  (%i1) solve(p^2+p+41=k*35,p);
                  sqrt(140 k - 163) + 1      sqrt(140 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(140*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        35
において、
(%i3) x:35*n;
(%o3)                                35 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                           1225 n  + 35 n
(%i5) x:35*n+1;
(%o5)                              35 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                         1225 n  + 105 n + 2
(%i7) x:35*n+2;
(%o7)                              35 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                         1225 n  + 175 n + 6
(%i9) x:35*n+3;
(%o9)                              35 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                       1225 n  + 245 n + 12
(%i11) x:35*n+4;
(%o11)                             35 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                       1225 n  + 315 n + 20
(%i13) x:35*n+5;
(%o13)                             35 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                       1225 n  + 385 n + 30
(%i15) x:35*n+6;
(%o15)                             35 n + 6
(%i16) expand(x*(x+1));
                                   2
(%o16)                       1225 n  + 455 n + 42
(%i17) mod(42,35);
(%o17)                                 7
(%i18) x:35*n+7;
(%o18)                             35 n + 7
(%i19) expand(x*(x+1));
                                   2
(%o19)                       1225 n  + 525 n + 56
(%i20) mod(56,35);
(%o20)                                21
(%i21) x:35*n+8;
(%o21)                             35 n + 8
(%i22) expand(x*(x+1));
                                   2
(%o22)                       1225 n  + 595 n + 72
(%i23) mod(72,35);
(%o23)                                 2
(%i24) x:35*n+9;
(%o24)                             35 n + 9
(%i25) expand(x*(x+1));
                                   2
(%o25)                       1225 n  + 665 n + 90
(%i26) mod(90,35);
(%o26)                                20
(%i27) x:35*n+10;
(%o27)                             35 n + 10
(%i28) expand(x*(x+1));
                                   2
(%o28)                       1225 n  + 735 n + 110
(%i29) mod(110,35);
(%o29)                                 5
(%i30) x:35*n+11;
(%o30)                             35 n + 11
(%i31) expand(x*(x+1));
                                   2
(%o31)                       1225 n  + 805 n + 132
(%i32) mod(132,35);
(%o32)                                27
(%i33) x:35*n+12;
(%o33)                             35 n + 12
(%i34) expand(x*(x+1));
                                   2
(%o34)                       1225 n  + 875 n + 156
(%i35) mod(156,35);
(%o35)                                16
(%i36) x:35*n+13;
(%o36)                             35 n + 13
(%i37) expand(x*(x+1));
                                   2
(%o37)                       1225 n  + 945 n + 182
(%i38) mod(182,35);
(%o38)                                 7
(%i39) x:35*n+14;
(%o39)                             35 n + 14
(%i40) expand(x*(x+1));
                                  2
(%o40)                      1225 n  + 1015 n + 210
(%i41) mod(210,35);
(%o41)                                 0
(%i42) x:35*n+15;
(%o42)                             35 n + 15
(%i43) expand(x*(x+1));
                                  2
(%o43)                      1225 n  + 1085 n + 240
(%i44) mod(240,35);
(%o44)                                30
(%i45) x:35*n+16;
(%o45)                             35 n + 16
(%i46) expand(x*(x+1));
                                  2
(%o46)                      1225 n  + 1155 n + 272
(%i47) mod(272,35);
(%o47)                                27
(%i48) x:35*n+17;
(%o48)                             35 n + 17
(%i49) expand(x*(x+1));
                                  2
(%o49)                      1225 n  + 1225 n + 306
(%i50) mod(306,35);
(%o50)                                26
(%i51) x:35*n+18;
(%o51)                             35 n + 18
(%i52) expand(x*(x+1));
                                  2
(%o52)                      1225 n  + 1295 n + 342
(%i53) mod(342,35);
(%o53)                                27
(%i54) x:35*n+19;
(%o54)                             35 n + 19
(%i55) expand(x*(x+1));
                                  2
(%o55)                      1225 n  + 1365 n + 380
(%i56) mod(380,35);
(%o56)                                30
(%i57) x:35*n+20;
(%o57)                             35 n + 20
(%i58) expand(x*(x+1));
                                  2
(%o58)                      1225 n  + 1435 n + 420
(%i59) mod(420,35);
(%o59)                                 0
(%i60) x:35*n+21;
(%o60)                             35 n + 21
(%i61) expand(x*(x+1));
                                  2
(%o61)                      1225 n  + 1505 n + 462
(%i62) mod(462,35);
(%o62)                                 7
(%i63) x:35*n+22;
(%o63)                             35 n + 22
(%i64) expand(x*(x+1));
                                  2
(%o64)                      1225 n  + 1575 n + 506
(%i65) mod(506,35);
(%o65)                                16
(%i66) x:35*n+23;
(%o66)                             35 n + 23
(%i67) expand(x*(x+1));
                                  2
(%o67)                      1225 n  + 1645 n + 552
(%i68) mod(552,35);
(%o68)                                27
(%i69) x:35*n+24;
(%o69)                             35 n + 24
(%i70) expand(x*(x+1));
                                  2
(%o70)                      1225 n  + 1715 n + 600
(%i71) mod(600,35);
(%o71)                                 5
(%i72) x:35*n+25;
(%o72)                             35 n + 25
(%i73) expand(x*(x+1));
                                  2
(%o73)                      1225 n  + 1785 n + 650
(%i74) mod(650,35);
(%o74)                                20
(%i75) x:35*n+26;
(%o75)                             35 n + 26
(%i76) expand(x*(x+1));
                                  2
(%o76)                      1225 n  + 1855 n + 702
(%i77) mod(702,35);
(%o77)                                 2
(%i78) x:35*n+27;
(%o78)                             35 n + 27
(%i79) expand(x*(x+1));
                                  2
(%o79)                      1225 n  + 1925 n + 756
(%i80) mod(756,35);
(%o80)                                21
(%i81) x:35*n+28;
(%o81)                             35 n + 28
(%i82) expand(x*(x+1));
                                  2
(%o82)                      1225 n  + 1995 n + 812
(%i83) mod(812,35);
(%o83)                                 7
(%i84) x:35*n+29;
(%o84)                             35 n + 29
(%i85) expand(x*(x+1));
                                  2
(%o85)                      1225 n  + 2065 n + 870
(%i86) mod(870,35);
(%o86)                                30
(%i87) x:35*n+30;
(%o87)                             35 n + 30
(%i88) expand(x*(x+1));
                                  2
(%o88)                      1225 n  + 2135 n + 930
(%i89) mod(930,35);
(%o89)                                20
(%i90) x:35*n+31;
(%o90)                             35 n + 31
(%i91) expand(x*(x+1));
                                  2
(%o91)                      1225 n  + 2205 n + 992
(%i92) mod(992,35);
(%o92)                                12
(%i93) x:35*n+32;
(%o93)                             35 n + 32
(%i94) expand(x*(x+1));
                                  2
(%o94)                      1225 n  + 2275 n + 1056
(%i95) mod(1056,35);
(%o95)                                 6
(%i96) x:35*n+33;
(%o96)                             35 n + 33
(%i97) expand(x*(x+1));
                                  2
(%o97)                      1225 n  + 2345 n + 1122
(%i98) mod(1122,35);
(%o98)                                 2
(%i99) x:35*n+34;
(%o99)                             35 n + 34
(%i100) expand(x*(x+1));
                                  2
(%o100)                     1225 n  + 2415 n + 1190
(%i101) mod(1190,35);
(%o101)                                0
(%i102) mod(41,35);
(%o102)                                6
mod(41,35)=6より、mod(x(x+1),35)=29でなければならない。しかし、mod(x(x+1),35)=0,2,5,6,7,11,12,16,17,20,21,23,24,26,27,30より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 6日(日)17時54分33秒
返信・引用
  (%i1) solve(p^2+p+41=k*33,p);
                  sqrt(132 k - 163) + 1      sqrt(132 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(132*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        33
において、
(%i3) x:33*n;
(%o3)                                33 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                           1089 n  + 33 n
(%i5) x:33*n+1;
(%o5)                              33 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                         1089 n  + 99 n + 2
(%i7) x:33*n+2;
(%o7)                              33 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                         1089 n  + 165 n + 6
(%i9) x:33*n+3;
(%o9)                              33 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                       1089 n  + 231 n + 12
(%i11) x:33*n+4;
(%o11)                             33 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                       1089 n  + 297 n + 20
(%i13) x:33*n+5;
(%o13)                             33 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                       1089 n  + 363 n + 30
(%i15) x:33*n+6;
(%o15)                             33 n + 6
(%i16) expand(x*(x+1));
                                   2
(%o16)                       1089 n  + 429 n + 42
(%i17) mod(42,33);
(%o17)                                 9
(%i18) x:33*n+7;
(%o18)                             33 n + 7
(%i19) expand(x*(x+1));
                                   2
(%o19)                       1089 n  + 495 n + 56
(%i20) mod(56,33);
(%o20)                                23
(%i21) x:33*n+8;
(%o21)                             33 n + 8
(%i22) expand(x*(x+1));
                                   2
(%o22)                       1089 n  + 561 n + 72
(%i23) mod(72,33);
(%o23)                                 6
(%i24) x:33*n+9;
(%o24)                             33 n + 9
(%i25) expand(x*(x+1));
                                   2
(%o25)                       1089 n  + 627 n + 90
(%i26) mod(90,33);
(%o26)                                24
(%i27) x:33*n+10;
(%o27)                             33 n + 10
(%i28) expand(x*(x+1));
                                   2
(%o28)                       1089 n  + 693 n + 110
(%i29) mod(110,33);
(%o29)                                11
(%i30) x:33*n+11;
(%o30)                             33 n + 11
(%i31) expand(x*(x+1));
                                   2
(%o31)                       1089 n  + 759 n + 132
(%i32) mod(132,33);
(%o32)                                 0
(%i33) x:33*n+12;
(%o33)                             33 n + 12
(%i34) expand(x*(x+1));
                                   2
(%o34)                       1089 n  + 825 n + 156
(%i35) mod(156,33);
(%o35)                                24
(%i36) x:33*n+13;
(%o36)                             33 n + 13
(%i37) expand(x*(x+1));
                                   2
(%o37)                       1089 n  + 891 n + 182
(%i38) mod(182,33);
(%o38)                                17
(%i39) x:33*n+14;
(%o39)                             33 n + 14
(%i40) expand(x*(x+1));
                                   2
(%o40)                       1089 n  + 957 n + 210
(%i41) mod(210,33);
(%o41)                                12
(%i42) x:33*n+15;
(%o42)                             33 n + 15
(%i43) expand(x*(x+1));
                                  2
(%o43)                      1089 n  + 1023 n + 240
(%i44) mod(240,33);
(%o44)                                 9
(%i45) x:33*n+16;
(%o45)                             33 n + 16
(%i46) expand(x*(x+1));
                                  2
(%o46)                      1089 n  + 1089 n + 272
(%i47) mod(272,33);
(%o47)                                 8
(%i48) x:33*n+17;
(%o48)                             33 n + 17
(%i49) expand(x*(x+1));
                                  2
(%o49)                      1089 n  + 1155 n + 306
(%i50) mod(306,33);
(%o50)                                 9
(%i51) x:33*n+18;
(%o51)                             33 n + 18
(%i52) expand(x*(x+1));
                                  2
(%o52)                      1089 n  + 1221 n + 342
(%i53) mod(342,33);
(%o53)                                12
(%i54) x:33*n+19;
(%o54)                             33 n + 19
(%i55) expand(x*(x+1));
                                  2
(%o55)                      1089 n  + 1287 n + 380
(%i56) mod(380,33);
(%o56)                                17
(%i57) x:33*n+20;
(%o57)                             33 n + 20
(%i58) expand(x*(x+1));
                                  2
(%o58)                      1089 n  + 1353 n + 420
(%i59) mod(420,33);
(%o59)                                24
(%i60) x:33*n+21;
(%o60)                             33 n + 21
(%i61) expand(x*(x+1));
                                  2
(%o61)                      1089 n  + 1419 n + 462
(%i62) mod(462,33);
(%o62)                                 0
(%i63) x:33*n+22;
(%o63)                             33 n + 22
(%i64) expand(x*(x+1));
                                  2
(%o64)                      1089 n  + 1485 n + 506
(%i65) mod(506,33);
(%o65)                                11
(%i66) x:33*n+23;
(%o66)                             33 n + 23
(%i67) expand(x*(x+1));
                                  2
(%o67)                      1089 n  + 1551 n + 552
(%i68) mod(552,33);
(%o68)                                24
(%i69) x:33*n+24;
(%o69)                             33 n + 24
(%i70) expand(x*(x+1));
                                  2
(%o70)                      1089 n  + 1617 n + 600
(%i71) mod(600,33);
(%o71)                                 6
(%i72) x:33*n+25;
(%o72)                             33 n + 25
(%i73) expand(x*(x+1));
                                  2
(%o73)                      1089 n  + 1683 n + 650
(%i74) mod(650,33);
(%o74)                                23
(%i75) x:33*n+26;
(%o75)                             33 n + 26
(%i76) expand(x*(x+1));
                                  2
(%o76)                      1089 n  + 1749 n + 702
(%i77) mod(702,33);
(%o77)                                 9
(%i78) x:33*n+27;
(%o78)                             33 n + 27
(%i79) expand(x*(x+1));
                                  2
(%o79)                      1089 n  + 1815 n + 756
(%i80) mod(756,33);
(%o80)                                30
(%i81) x:33*n+28;
(%o81)                             33 n + 28
(%i82) expand(x*(x+1));
                                  2
(%o82)                      1089 n  + 1881 n + 812
(%i83) mod(812,33);
(%o83)                                20
(%i84) x:33*n+29;
(%o84)                             33 n + 29
(%i85) expand(x*(x+1));
                                  2
(%o85)                      1089 n  + 1947 n + 870
(%i86) mod(870,33);
(%o86)                                12
(%i87) x:33*n+30;
(%o87)                             33 n + 30
(%i88) expand(x*(x+1));
                                  2
(%o88)                      1089 n  + 2013 n + 930
(%i89) mod(930,33);
(%o89)                                 6
(%i90) x:33*n+31;
(%o90)                             33 n + 31
(%i91) expand(x*(x+1));
                                  2
(%o91)                      1089 n  + 2079 n + 992
(%i92) mod(992,33);
(%o92)                                 2
(%i93) x:33*n+32;
(%o93)                             33 n + 32
(%i94) expand(x*(x+1));
                                  2
(%o94)                      1089 n  + 2145 n + 1056
(%i95) mod(1056,33);
(%o95)                                 0
mod(41,33)=8より、mod(x(x+1),33)=25でなければならない。しかし、mod(x(x+1),33)=0,1,2,6,8,9,11,12,17,20,23,24,30より、kは整数にならない。
 

Re: 有理数と無理数

 投稿者:うんざりはちべえ  投稿日:2018年 5月 6日(日)13時59分49秒
返信・引用
  有理数は4則において閉じているので、
ライプニッツの公式https://ja.wikipedia.org/wiki/%E3%83%A9%E3%82%A4%E3%83%97%E3%83%8B%E3%83%83%E3%83%84%E3%81%AE%E5%85%AC%E5%BC%8F
は、誤りである。なぜなら、左辺は有理数右辺は無理数であるから等号で結ぶことはできない。
 

Re: 有理数と無理数

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)19時54分45秒
返信・引用
  有理数と無理数は実数の集合で、有理数の集合を作ると無理数の集合は有理数の集合の補集合である。つまり、有理数でないものは無理数である。また、無理数でないものは有理数である。

さて、有理数の進数があるのだから、無理数の進数もある。
無理数の進数では有理数とその補集合として無理数がある。

すると、有理数と無理数とは一体何であろうかというところに行き着く。

そこまで行き着けば、デデキントの切断は、正しいのであろうかというところに行き着く。
 

有理数と無理数

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)19時25分42秒
返信・引用
  有理数の集合では、
有理数+有理数=有理数
有理数ー有理数=有理数
有理数X有理数=有理数
有理数÷有理数=有理数
であり、閉じている。

無理数の集合では、
無理数+無理数=無理数
無理数ー無理数=無理数あるいは有理数√2ー√2=0
無理数x無理数=無理数あるいは有理数√2x√2=2
無理数÷無理数=無理数あるいは有理数√2÷√2=1
のように閉じていない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)19時09分46秒
返信・引用 編集済
  (%i1) solve(p^2+p+41=k*31,p);
                  sqrt(124 k - 163) + 1      sqrt(124 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(124*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        31
において、
(%i3) x:31*n;
(%o3)                                31 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            961 n  + 31 n
(%i5) x:31*n+1;
(%o5)                              31 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          961 n  + 93 n + 2
(%i7) x:31*n+2;
(%o7)                              31 n + 2
(%i8) expand(x*(x+1));
                                   2
(%o8)                         961 n  + 155 n + 6
(%i9) x:31*n+3;
(%o9)                              31 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                        961 n  + 217 n + 12
(%i11) x:31*n+4;
(%o11)                             31 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                        961 n  + 279 n + 20
(%i13) x:31*n+5;
(%o13)                             31 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                        961 n  + 341 n + 30
(%i15) x:31*n+6;
(%o15)                             31 n + 6
(%i16) expand(x*(x+1));
                                   2
(%o16)                        961 n  + 403 n + 42
(%i17) mod(42,31);
(%o17)                                11
(%i18) x:31*n+7;
(%o18)                             31 n + 7
(%i19) expand(x*(x+1));
                                   2
(%o19)                        961 n  + 465 n + 56
(%i20) mod(56,31);
(%o20)                                25
(%i21) x:31*n+8;
(%o21)                             31 n + 8
(%i22) expand(x*(x+1));
                                   2
(%o22)                        961 n  + 527 n + 72
(%i23) mod(72,31);
(%o23)                                10
(%i24) x:31*n+9;
(%o24)                             31 n + 9
(%i25) expand(x*(x+1));
                                   2
(%o25)                        961 n  + 589 n + 90
(%i26) mod(90,31);
(%o26)                                28
(%i27) x:31*n+10;
(%o27)                             31 n + 10
(%i28) expand(x*(x+1));
                                  2
(%o28)                       961 n  + 651 n + 110
(%i29) mod(110,31);
(%o29)                                17
(%i30) x:31*n+11;
(%o30)                             31 n + 11
(%i31) expand(x*(x+1));
                                  2
(%o31)                       961 n  + 713 n + 132
(%i32) mod(132,31);
(%o32)                                 8
(%i33) x:31*n+12;
(%o33)                             31 n + 12
(%i34) expand(x*(x+1));
                                  2
(%o34)                       961 n  + 775 n + 156
(%i35) mod(156,31);
(%o35)                                 1
(%i36) x:31*n+13;
(%o36)                             31 n + 13
(%i37) expand(x*(x+1));
                                  2
(%o37)                       961 n  + 837 n + 182
(%i38) mod(182,31);
(%o38)                                27
(%i39) x:31*n+14;
(%o39)                             31 n + 14
(%i40) expand(x*(x+1));
                                  2
(%o40)                       961 n  + 899 n + 210
(%i41) mod(210,31);
(%o41)                                24
(%i42) x:31*n+15;
(%o42)                             31 n + 15
(%i43) expand(x*(x+1));
                                  2
(%o43)                       961 n  + 961 n + 240
(%i44) mod(240,31);
(%o44)                                23
(%i45) x:31*n+16;
(%o45)                             31 n + 16
(%i46) expand(x*(x+1));
                                  2
(%o46)                       961 n  + 1023 n + 272
(%i47) mod(272,31);
(%o47)                                24
(%i48) x:31*n+17;
(%o48)                             31 n + 17
(%i49) expand(x*(x+1));
                                  2
(%o49)                       961 n  + 1085 n + 306
(%i50) mod(306,31);
(%o50)                                27
(%i51) x:31*n+18;
(%o51)                             31 n + 18
(%i52) expand(x*(x+1));
                                  2
(%o52)                       961 n  + 1147 n + 342
(%i53) mod(342,31);
(%o53)                                 1
(%i54) x:31*n+19;
(%o54)                             31 n + 19
(%i55) expand(x*(x+1));
                                  2
(%o55)                       961 n  + 1209 n + 380
(%i56) mod(380,31);
(%o56)                                 8
(%i57) x:31*n+20;
(%o57)                             31 n + 20
(%i58) expand(x*(x+1));
                                  2
(%o58)                       961 n  + 1271 n + 420
(%i59) mod(420,31);
(%o59)                                17
(%i60) x:31*n+21;
(%o60)                             31 n + 21
(%i61) expand(x*(x+1));
                                  2
(%o61)                       961 n  + 1333 n + 462
(%i62) mod(462,31);
(%o62)                                28
(%i63) x:31*n+22;
(%o63)                             31 n + 22
(%i64) expand(x*(x+1));
                                  2
(%o64)                       961 n  + 1395 n + 506
(%i65) mod(506,31);
(%o65)                                10
(%i66) x:31*n+23;
(%o66)                             31 n + 23
(%i67) expand(x*(x+1));
                                  2
(%o67)                       961 n  + 1457 n + 552
(%i68) mod(552,31);
(%o68)                                25
(%i69) x:31*n+24;
(%o69)                             31 n + 24
(%i70) expand(x*(x+1));
                                  2
(%o70)                       961 n  + 1519 n + 600
(%i71) mod(600,31);
(%o71)                                11
(%i72) x:31*n+25;
(%o72)                             31 n + 25
(%i73) expand(x*(x+1));
                                  2
(%o73)                       961 n  + 1581 n + 650
(%i74) mod(650,31);
(%o74)                                30
(%i75) x:31*n+26;
(%o75)                             31 n + 26
(%i76) expand(x*(x+1));
                                  2
(%o76)                       961 n  + 1643 n + 702
(%i77) mod(702,31);
(%o77)                                20
(%i78) x:31*n+27;
(%o78)                             31 n + 27
(%i79) expand(x*(x+1));
                                  2
(%o79)                       961 n  + 1705 n + 756
(%i80) mod(756,31);
(%o80)                                12
(%i81) x:31*n+28;
(%o81)                             31 n + 28
(%i82) expand(x*(x+1));
                                  2
(%o82)                       961 n  + 1767 n + 812
(%i83) mod(812,31);
(%o83)                                 6
(%i84) x:31*n+29;
(%o84)                             31 n + 29
(%i85) expand(x*(x+1));
                                  2
(%o85)                       961 n  + 1829 n + 870
(%i86) mod(870,31);
(%o86)                                 2
(%i87) x:31*n+30;
(%o87)                             31 n + 30
(%i88) expand(x*(x+1));
                                  2
(%o88)                       961 n  + 1891 n + 930
(%i89) mod(930,31);
(%o89)                                 0
mod(41,31)=10より、mod(x(x+1),31)=21でなければならない。しかし、mod(x(x+1),31)=0,1,2,6,7,8,10,11,12,17,20,23,24,25,27,28,30より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時31分45秒
返信・引用
  (%i3) solve(p^2+p+41=k*29,p);
                  sqrt(116 k - 163) + 1      sqrt(116 k - 163) - 1
(%o3)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i4) solve(116*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o4)                          [k = -----------]
                                        29
において、
(%i5) x:29*n;
(%o5)                                29 n
(%i6) expand(x*(x+1));
                                      2
(%o6)                            841 n  + 29 n
(%i7) x:29*n+1;
(%o7)                              29 n + 1
(%i8) expand(x*(x+1));
                                    2
(%o8)                          841 n  + 87 n + 2
(%i9) x:29*n+2;
(%o9)                              29 n + 2
(%i10) expand(x*(x+1));
                                   2
(%o10)                        841 n  + 145 n + 6
(%i11) x:29*n+3;
(%o11)                             29 n + 3
(%i12) expand(x*(x+1));
                                   2
(%o12)                        841 n  + 203 n + 12
(%i13) x:29*n+4;
(%o13)                             29 n + 4
(%i14) expand(x*(x+1));
                                   2
(%o14)                        841 n  + 261 n + 20
(%i15) x:29*n+5;
(%o15)                             29 n + 5
(%i16) expand(x*(x+1));
                                   2
(%o16)                        841 n  + 319 n + 30
(%i17) mod(30,29);
(%o17)                                 1
(%i18) x:29*n+6;
(%o18)                             29 n + 6
(%i19) expand(x*(x+1));
                                   2
(%o19)                        841 n  + 377 n + 42
(%i20) mod(42,29);
(%o20)                                13
(%i21) x:29*n+7;
(%o21)                             29 n + 7
(%i22) expand(x*(x+1));
                                   2
(%o22)                        841 n  + 435 n + 56
(%i23) mod(56,29);
(%o23)                                27
(%i24) x:29*n+8;
(%o24)                             29 n + 8
(%i25) expand(x*(x+1));
                                   2
(%o25)                        841 n  + 493 n + 72
(%i26) mod(72,29);
(%o26)                                14
(%i27) x:29*n+9;
(%o27)                             29 n + 9
(%i28) expand(x*(x+1));
                                   2
(%o28)                        841 n  + 551 n + 90
(%i29) mod(90,29);
(%o29)                                 3
(%i30) x:29*n+10;
(%o30)                             29 n + 10
(%i31) expand(x*(x+1));
                                  2
(%o31)                       841 n  + 609 n + 110
(%i32) mod(110,29);
(%o32)                                23
(%i33) x:29*n+11;
(%o33)                             29 n + 11
(%i34) expand(x*(x+1));
                                  2
(%o34)                       841 n  + 667 n + 132
(%i35) mod(132,29);
(%o35)                                16
(%i36) x:29*n+12;
(%o36)                             29 n + 12
(%i37) expand(x*(x+1));
                                  2
(%o37)                       841 n  + 725 n + 156
(%i38) mod(156,29);
(%o38)                                11
(%i39) x:29*n+13;
(%o39)                             29 n + 13
(%i40) expand(x*(x+1));
                                  2
(%o40)                       841 n  + 783 n + 182
(%i41) mod(182,29);
(%o41)                                 8
(%i42) x:29*n+14;
(%o42)                             29 n + 14
(%i43) expand(x*(x+1));
                                  2
(%o43)                       841 n  + 841 n + 210
(%i44) mod(210,29);
(%o44)                                 7
(%i45) x:29*n+15;
(%o45)                             29 n + 15
(%i46) expand(x*(x+1));
                                  2
(%o46)                       841 n  + 899 n + 240
(%i47) mod(240,29);
(%o47)                                 8
(%i48) x:29*n+16;
(%o48)                             29 n + 16
(%i49) expand(x*(x+1));
                                  2
(%o49)                       841 n  + 957 n + 272
(%i50) mod(272,29);
(%o50)                                11
(%i51) x:29*n+17;
(%o51)                             29 n + 17
(%i52) expand(x*(x+1));
                                  2
(%o52)                       841 n  + 1015 n + 306
(%i53) mod(306,29);
(%o53)                                16
(%i54) x:29*n+18;
(%o54)                             29 n + 18
(%i55) expand(x*(x+1));
                                  2
(%o55)                       841 n  + 1073 n + 342
(%i56) mod(342,29);
(%o56)                                23
(%i57) x:29*n+19;
(%o57)                             29 n + 19
(%i58) expand(x*(x+1));
                                  2
(%o58)                       841 n  + 1131 n + 380
(%i59) mod(380,29);
(%o59)                                 3
(%i60) x:29*n+20;
(%o60)                             29 n + 20
(%i61) expand(x*(x+1));
                                  2
(%o61)                       841 n  + 1189 n + 420
(%i62) mod(420,29);
(%o62)                                14
(%i63) x:29*n+21;
(%o63)                             29 n + 21
(%i64) expand(x*(x+1));
                                  2
(%o64)                       841 n  + 1247 n + 462
(%i65) mod(462,29);
(%o65)                                27
(%i66) x:29*n+22;
(%o66)                             29 n + 22
(%i67) expand(x*(x+1));
                                  2
(%o67)                       841 n  + 1305 n + 506
(%i68) mod(506,29);
(%o68)                                13
(%i69) x:29*n+23;
(%o69)                             29 n + 23
(%i70) expand(x*(x+1));
                                  2
(%o70)                       841 n  + 1363 n + 552
(%i71) mod(552,29);
(%o71)                                 1
(%i72) x:29*n+24;
(%o72)                             29 n + 24
(%i73) expand(x*(x+1));
                                  2
(%o73)                       841 n  + 1421 n + 600
(%i74) mod(600,29);
(%o74)                                20
(%i75) x:29*n+25;
(%o75)                             29 n + 25
(%i76) expand(x*(x+1));
                                  2
(%o76)                       841 n  + 1479 n + 650
(%i77) mod(650,29);
(%o77)                                12
(%i78) x:29*n+26;
(%o78)                             29 n + 26
(%i79) expand(x*(x+1));
                                  2
(%o79)                       841 n  + 1537 n + 702
(%i80) mod(702,29);
(%o80)                                 6
(%i81) x:29*n+27;
(%o81)                             29 n + 27
(%i82) expand(x*(x+1));
                                  2
(%o82)                       841 n  + 1595 n + 756
(%i83) mod(756,29);
(%o83)                                 2
(%i84) x:29*n+28;
(%o84)                             29 n + 28
(%i85) expand(x*(x+1));
                                  2
(%o85)                       841 n  + 1653 n + 812
(%i86) mod(812,29);
(%o86)                                 0
mod(41,29)=12より、mod(x(x+1),29)=17でなければならない。しかしmod(x(x+1),29)=0,1,2,3,6,7,8,11,12,13,14,16,20,23,27より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時31分0秒
返信・引用
  (%i1) solve(p^2+p+41=k*27,p);
                  sqrt(108 k - 163) + 1      sqrt(108 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(108*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        27
において、
(%i3) x:27*n;
(%o3)                                27 n
(%i7) expand(x*(x+1));
                                      2
(%o7)                            729 n  + 27 n
(%i4) x:27*n+1;
(%o4)                              27 n + 1
(%i5) expand(x*(x+1));
                                    2
(%o5)                          729 n  + 81 n + 2
(%i8) x:27*n+2;
(%o8)                              27 n + 2
(%i9) expand(x*(x+1));
                                   2
(%o9)                         729 n  + 135 n + 6
(%i10) x:27*n+3;
(%o10)                             27 n + 3
(%i11) expand(x*(x+1));
                                   2
(%o11)                        729 n  + 189 n + 12
(%i12) x:27*n+4;
(%o12)                             27 n + 4
(%i13) expand(x*(x+1));
                                   2
(%o13)                        729 n  + 243 n + 20
(%i14) x:27*n+5;
(%o14)                             27 n + 5
(%i15) expand(x*(x+1));
                                   2
(%o15)                        729 n  + 297 n + 30
(%i16) mod(30,27);
(%o16)                                 3
(%i17) x:27*n+6;
(%o17)                             27 n + 6
(%i18) expand(x*(x+1));
                                   2
(%o18)                        729 n  + 351 n + 42
(%i19) mod(42,27);
(%o19)                                15
(%i20) x:27*n+7;
(%o20)                             27 n + 7
(%i21) expand(x*(x+1));
                                   2
(%o21)                        729 n  + 405 n + 56
(%i22) mod(56,27);
(%o22)                                 2
(%i23) x:27*n+8;
(%o23)                             27 n + 8
(%i24) expand(x*(x+1));
                                   2
(%o24)                        729 n  + 459 n + 72
(%i25) mod(72,27);
(%o25)                                18
(%i26) x:27*n+9;
(%o26)                             27 n + 9
(%i27) expand(x*(x+1));
                                   2
(%o27)                        729 n  + 513 n + 90
(%i28) mod(90,27);
(%o28)                                 9
(%i29) x:27*n+10;
(%o29)                             27 n + 10
(%i30) expand(x*(x+1));
                                  2
(%o30)                       729 n  + 567 n + 110
(%i31) mod(110,27);
(%o31)                                 2
(%i32) x:27*n+11;
(%o32)                             27 n + 11
(%i33) expand(x*(x+1));
                                  2
(%o33)                       729 n  + 621 n + 132
(%i34) mod(132,27);
(%o34)                                24
(%i35) x:27*n+12;
(%o35)                             27 n + 12
(%i36) expand(x*(x+1));
                                  2
(%o36)                       729 n  + 675 n + 156
(%i37) mod(156,27);
(%o37)                                21
(%i38) x:27*n+13;
(%o38)                             27 n + 13
(%i39) expand(x*(x+1));
                                  2
(%o39)                       729 n  + 729 n + 182
(%i40) mod(182,27);
(%o40)                                20
(%i41) x:27*n+14;
(%o41)                             27 n + 14
(%i42) expand(x*(x+1));
                                  2
(%o42)                       729 n  + 783 n + 210
(%i43) mod(210,27);
(%o43)                                21
(%i44) x:27*n+15;
(%o44)                             27 n + 15
(%i45) expand(x*(x+1));
                                  2
(%o45)                       729 n  + 837 n + 240
(%i46) mod(240,27);
(%o46)                                24
(%i47) x:27*n+16;
(%o47)                             27 n + 16
(%i48) expand(x*(x+1));
                                  2
(%o48)                       729 n  + 891 n + 272
(%i49) mod(272,27);
(%o49)                                 2
(%i50) x:27*n+17;
(%o50)                             27 n + 17
(%i51) expand(x*(x+1));
                                  2
(%o51)                       729 n  + 945 n + 306
(%i52) mod(306,27);
(%o52)                                 9
(%i53) x:27*n+18;
(%o53)                             27 n + 18
(%i54) expand(x*(x+1));
                                  2
(%o54)                       729 n  + 999 n + 342
(%i55) mod(342,27);
(%o55)                                18
(%i56) x:27*n+19;
(%o56)                             27 n + 19
(%i57) expand(x*(x+1));
                                  2
(%o57)                       729 n  + 1053 n + 380
(%i58) mod(380,27);
(%o58)                                 2
(%i59) x:27*n+20;
(%o59)                             27 n + 20
(%i60) expand(x*(x+1));
                                  2
(%o60)                       729 n  + 1107 n + 420
(%i61) mod(420,27);
(%o61)                                15
(%i62) x:27*n+21;
(%o62)                             27 n + 21
(%i63) expand(x*(x+1));
                                  2
(%o63)                       729 n  + 1161 n + 462
(%i64) mod(462,27);
(%o64)                                 3
(%i65) x:27*n+22;
(%o65)                             27 n + 22
(%i66) expand(x*(x+1));
                                  2
(%o66)                       729 n  + 1215 n + 506
(%i67) mod(506,27);
(%o67)                                20
(%i68) x:27*n+23;
(%o68)                             27 n + 23
(%i69) expand(x*(x+1));
                                  2
(%o69)                       729 n  + 1269 n + 552
(%i70) mod(552,27);
(%o70)                                12
(%i71) x:27*n+24;
(%o71)                             27 n + 24
(%i72) expand(x*(x+1));
                                  2
(%o72)                       729 n  + 1323 n + 600
(%i73) mod(600,27);
(%o73)                                 6
(%i74) x:27*n+25;
(%o74)                             27 n + 25
(%i75) expand(x*(x+1));
                                  2
(%o75)                       729 n  + 1377 n + 650
(%i76) mod(650,27);
(%o76)                                 2
(%i77) x:27*n+26;
(%o77)                             27 n + 26
(%i78) expand(x*(x+1));
                                  2
(%o78)                       729 n  + 1431 n + 702
(%i79) mod(702,27);
(%o79)                                 0
(%i80) mod(41,27);
(%o80)                                14
(%i81)

mod(41,27)=14より、mod(x(x+1),27)=13でなければならない。しかしmod(x(x+1),27)=0,2,3,6,9,12,15,18,20,21,24より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時30分15秒
返信・引用
  (%i1) solve(p^2+p+41=k*25,p);
                  sqrt(100 k - 163) + 1      sqrt(100 k - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
より、
(%i2) solve(100*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        25
において、
(%i3) x:25*n;
(%o3)                                25 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            625 n  + 25 n
(%i5) x:25*n+1;
(%o5)                              25 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          625 n  + 75 n + 2
(%i7) x:25*n+2;
(%o7)                              25 n + 2
(%i8) expand(x*(x+1));
                                   2
(%o8)                         625 n  + 125 n + 6
(%i9) x:25*n+3;
(%o9)                              25 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                        625 n  + 175 n + 12
(%i11) x:25*n+4;
(%o11)                             25 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                        625 n  + 225 n + 20
(%i13) x:25*n+5;
(%o13)                             25 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                        625 n  + 275 n + 30
(%i15) mod(30,25);
(%o15)                                 5
(%i16) x:25*n+6;
(%o16)                             25 n + 6
(%i17) expand(x*(x+1));
                                   2
(%o17)                        625 n  + 325 n + 42
(%i18) mod(42,25);
(%o18)                                17
(%i19) x:25*n+7;
(%o19)                             25 n + 7
(%i20) expand(x*(x+1));
                                   2
(%o20)                        625 n  + 375 n + 56
(%i21) mod(56,25);
(%o21)                                 6
(%i22) x:25*n+8;
(%o22)                             25 n + 8
(%i23) expand(x*(x+1));
                                   2
(%o23)                        625 n  + 425 n + 72
(%i24) mod(72,25);
(%o24)                                22
(%i25) x:25*n+9;
(%o25)                             25 n + 9
(%i26) expand(x*(x+1));
                                   2
(%o26)                        625 n  + 475 n + 90
(%i27) mod(90,25);
(%o27)                                15
(%i28) x:25*n+10;
(%o28)                             25 n + 10
(%i29) expand(x*(x+1));
                                  2
(%o29)                       625 n  + 525 n + 110
(%i30) mod(110,25);
(%o30)                                10
(%i31) x:25*n+11;
(%o31)                             25 n + 11
(%i32) expand(x*(x+1));
                                  2
(%o32)                       625 n  + 575 n + 132
(%i33) mod(132,25);
(%o33)                                 7
(%i34) x:25*n+12;
(%o34)                             25 n + 12
(%i35) expand(x*(x+1));
                                  2
(%o35)                       625 n  + 625 n + 156
(%i36) mod(156,25);
(%o36)                                 6
(%i37) x:25*n+13;
(%o37)                             25 n + 13
(%i38) expand(x*(x+1));
                                  2
(%o38)                       625 n  + 675 n + 182
(%i39) mod(182,25);
(%o39)                                 7
(%i40) x:25*n+14;
(%o40)                             25 n + 14
(%i41) expand(x*(x+1));
                                  2
(%o41)                       625 n  + 725 n + 210
(%i42) mod(210,25);
(%o42)                                10
(%i43) x:25*n+15;
(%o43)                             25 n + 15
(%i44) expand(x*(x+1));
                                  2
(%o44)                       625 n  + 775 n + 240
(%i45) mod(240,25);
(%o45)                                15
(%i46) x:25*n+16;
(%o46)                             25 n + 16
(%i47) expand(x*(x+1));
                                  2
(%o47)                       625 n  + 825 n + 272
(%i48) mod(272,25);
(%o48)                                22
(%i49) x:25*n+17;
(%o49)                             25 n + 17
(%i50) expand(x*(x+1));
                                  2
(%o50)                       625 n  + 875 n + 306
(%i51) mod(306,25);
(%o51)                                 6
(%i52) x:25*n+18;
(%o52)                             25 n + 18
(%i53) expand(x*(x+1));
                                  2
(%o53)                       625 n  + 925 n + 342
(%i54) mod(342,25);
(%o54)                                17
(%i55) x:25*n+19;
(%o55)                             25 n + 19
(%i56) expand(x*(x+1));
                                  2
(%o56)                       625 n  + 975 n + 380
(%i57) mod(380,25);
(%o57)                                 5
(%i58) x:25*n+20;
(%o58)                             25 n + 20
(%i59) expand(x*(x+1));
                                  2
(%o59)                       625 n  + 1025 n + 420
(%i60) mod(420,25);
(%o60)                                20
(%i61) x:25*n+21;
(%o61)                             25 n + 21
(%i62) expand(x*(x+1));
                                  2
(%o62)                       625 n  + 1075 n + 462
(%i63) mod(462,25);
(%o63)                                12
(%i64) x:25*n+22;
(%o64)                             25 n + 22
(%i65) expand(x*(x+1));
                                  2
(%o65)                       625 n  + 1125 n + 506
(%i66) mod(506,25);
(%o66)                                 6
(%i67) x:25*n+23;
(%o67)                             25 n + 23
(%i68) expand(x*(x+1));
                                  2
(%o68)                       625 n  + 1175 n + 552
(%i69) mod(552,25);
(%o69)                                 2
(%i70) x:25*n+24;
(%o70)                             25 n + 24
(%i71) expand(x*(x+1));
                                  2
(%o71)                       625 n  + 1225 n + 600
(%i72) mod(600,25);
(%o72)                                 0
(%i73)

mod(41,25)=16より、mod(x(x+1),25)=9でなければならない。しかしmod(x(x+1),25)=0,2,3,5.6,7,10,12,15,17,20,22より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時29分15秒
返信・引用
  (%i1) solve(p^2+p+41=k*23,p);
                   sqrt(92 k - 163) + 1      sqrt(92 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i2) solve(92*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        23
において、
(%i3) x:23*n;
(%o3)                                23 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            529 n  + 23 n
(%i5) x:23*n+1;
(%o5)                              23 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          529 n  + 69 n + 2
(%i7) x:23*n+2;
(%o7)                              23 n + 2
(%i8) expand(x*(x+1));
                                   2
(%o8)                         529 n  + 115 n + 6
(%i10) x:23*n+3;
(%o10)                             23 n + 3
(%i11) expand(x*(x+1));
                                   2
(%o11)                        529 n  + 161 n + 12
(%i12) x:23*n+4;
(%o12)                             23 n + 4
(%i13) expand(x*(x+1));
                                   2
(%o13)                        529 n  + 207 n + 20
(%i14) x:23*n+5;
(%o14)                             23 n + 5
(%i15) expand(x*(x+1));
                                   2
(%o15)                        529 n  + 253 n + 30
(%i16) mod(30,23);
(%o16)                                 7
(%i17) x:23*n+6;
(%o17)                             23 n + 6
(%i18) expand(x*(x+1));
                                   2
(%o18)                        529 n  + 299 n + 42
(%i19) mod(42,23);
(%o19)                                19
(%i20) x:23*n+7;
(%o20)                             23 n + 7
(%i21) expand(x*(x+1));
                                   2
(%o21)                        529 n  + 345 n + 56
(%i22) mod(56,23);
(%o22)                                10
(%i23) x:23*n+8;
(%o23)                             23 n + 8
(%i24) expand(x*(x+1));
                                   2
(%o24)                        529 n  + 391 n + 72
(%i25) mod(72,23);
(%o25)                                 3
(%i26) x:23*n+9;
(%o26)                             23 n + 9
(%i27) expand(x*(x+1));
                                   2
(%o27)                        529 n  + 437 n + 90
(%i28) mod(80,23);
(%o28)                                11
(%i29) x:23*n+10;
(%o29)                             23 n + 10
(%i30) expand(x*(x+1));
                                  2
(%o30)                       529 n  + 483 n + 110
(%i31) mod(110,23);
(%o31)                                18
(%i32) x:23*n+11;
(%o32)                             23 n + 11
(%i33) expand(x*(x+1));
                                  2
(%o33)                       529 n  + 529 n + 132
(%i34) mod(132,23);
(%o34)                                17
(%i35) x:23*n+12;
(%o35)                             23 n + 12
(%i36) expand(x*(x+1));
                                  2
(%o36)                       529 n  + 575 n + 156
(%i37) mod(156,23);
(%o37)                                18
(%i38) x:23*n+13;
(%o38)                             23 n + 13
(%i39) expand(x*(x+1));
                                  2
(%o39)                       529 n  + 621 n + 182
(%i40) mod(182,23);
(%o40)                                21
(%i41) x:23*n+14;
(%o41)                             23 n + 14
(%i42) expand(x*(x+1));
                                  2
(%o42)                       529 n  + 667 n + 210
(%i43) mod(210,23);
(%o43)                                 3
(%i44) x:23*n+15;
(%o44)                             23 n + 15
(%i45) expand(x*(x+1));
                                  2
(%o45)                       529 n  + 713 n + 240
(%i46) mod(240,23);
(%o46)                                10
(%i47) x:23*n+16;
(%o47)                             23 n + 16
(%i48) expand(x*(x+1));
                                  2
(%o48)                       529 n  + 759 n + 272
(%i49) mod(272,23);
(%o49)                                19
(%i50) x:23*n+17;
(%o50)                             23 n + 17
(%i51) expand(x*(x+1));
                                  2
(%o51)                       529 n  + 805 n + 306
(%i52) mod(306,23);
(%o52)                                 7
(%i53) x:23*n+18;
(%o53)                             23 n + 18
(%i54) expand(x*(x+1));
                                  2
(%o54)                       529 n  + 851 n + 342
(%i55) mod(342,23);
(%o55)                                20
(%i56) x:23*n+19;
(%o56)                             23 n + 19
(%i57) expand(x*(x+1));
                                  2
(%o57)                       529 n  + 897 n + 380
(%i58) mod(380,23);
(%o58)                                12
(%i59) x:23*n+20;
(%o59)                             23 n + 20
(%i60) expand(x*(x+1));
                                  2
(%o60)                       529 n  + 943 n + 420
(%i61) mod(420,23);
(%o61)                                 6
(%i62) x:23*n+21;
(%o62)                             23 n + 21
(%i63) expand(x*(x+1));
                                  2
(%o63)                       529 n  + 989 n + 462
(%i64) mod(462,23);
(%o64)                                 2
(%i65) x:23*n+22;
(%o65)                             23 n + 22
(%i66) expand(x*(x+1));
                                  2
(%o66)                       529 n  + 1035 n + 506
(%i67) mod(506,23);
(%o67)                                 0
(%i68)
mod(41,23)=18より、mod(x(x+1),23)=5でなければならない。しかしmod(x(x+1),23)=0,2,3,6,7,9,10,11,12,17,18,19,20,21より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時27分30秒
返信・引用
  (%i1) solve(p^2+p+41=k*21,p);
                   sqrt(84 k - 163) + 1      sqrt(84 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i2) solve(84*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        21
において、
(%i3) x:21*n;
(%o3)                                21 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            441 n  + 21 n
(%i5) x:21*n+1;
(%o5)                              21 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          441 n  + 63 n + 2
(%i7) x:21*n+2;
(%o7)                              21 n + 2
(%i8) expand(x*(x+1));
                                   2
(%o8)                         441 n  + 105 n + 6
(%i9) x:21*n+3;
(%o9)                              21 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                        441 n  + 147 n + 12
(%i11) x:21*n+4;
(%o11)                             21 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                        441 n  + 189 n + 20
(%i13) x:21*n+5;
(%o13)                             21 n + 5
(%i14) expand(x*(x+1));
                                   2
(%o14)                        441 n  + 231 n + 30
(%i15) mod(30,21);
(%o15)                                 9
(%i16) x:21*n+6;
(%o16)                             21 n + 6
(%i17) expand(x*(x+1));
                                   2
(%o17)                        441 n  + 273 n + 42
(%i18) mod(42,21);
(%o18)                                 0
(%i19) x:21*n+7;
(%o19)                             21 n + 7
(%i20) expand(x*(x+1));
                                   2
(%o20)                        441 n  + 315 n + 56
(%i21) mod(56,21);
(%o21)                                14
(%i22) x:21*n+8;
(%o22)                             21 n + 8
(%i23) expand(x*(x+1));
                                   2
(%o23)                        441 n  + 357 n + 72
(%i24) mod(72,21);
(%o24)                                 9
(%i25) x:21*n+9;
(%o25)                             21 n + 9
(%i26) expand(x*(x+1));
                                   2
(%o26)                        441 n  + 399 n + 90
(%i27) mod(90,21);
(%o27)                                 6
(%i28) x:21*n+10;
(%o28)                             21 n + 10
(%i29) expand(x*(x+1));
                                  2
(%o29)                       441 n  + 441 n + 110
(%i30) mod(110,21);
(%o30)                                 5
(%i31) x:21*n+11;
(%o31)                             21 n + 11
(%i32) expand(x*(x+1));
                                  2
(%o32)                       441 n  + 483 n + 132
(%i33) mod(132,21);
(%o33)                                 6
(%i34) x:21*n+12;
(%o34)                             21 n + 12
(%i35) expand(x*(x+1));
                                  2
(%o35)                       441 n  + 525 n + 156
(%i36) mod(156,21);
(%o36)                                 9
(%i37) x:21*n+13;
(%o37)                             21 n + 13
(%i38) expand(x*(x+1));
                                  2
(%o38)                       441 n  + 567 n + 182
(%i39) mod(182,21);
(%o39)                                14
(%i40) x:21*n+14;
(%o40)                             21 n + 14
(%i41) expand(x*(x+1));
                                  2
(%o41)                       441 n  + 609 n + 210
(%i42) mod(210,21);
(%o42)                                 0
(%i43) x:21*n+15;
(%o43)                             21 n + 15
(%i44) expand(x*(x+1));
                                  2
(%o44)                       441 n  + 651 n + 240
(%i45) mod(240,21);
(%o45)                                 9
(%i46) x:21*n+16;
(%o46)                             21 n + 16
(%i47) expand(x*(x+1));
                                  2
(%o47)                       441 n  + 693 n + 272
(%i48) mod(272,21);
(%o48)                                20
(%i49) x:21*n+17;
(%o49)                             21 n + 17
(%i50) expand(x*(x+1));
                                  2
(%o50)                       441 n  + 735 n + 306
(%i51) mod(306,21);
(%o51)                                12
(%i52) x:21*n+18;
(%o52)                             21 n + 18
(%i53) expand(x*(x+1));
                                  2
(%o53)                       441 n  + 777 n + 342
(%i54) mod(342,21);
(%o54)                                 6
(%i55) x:21*n+19;
(%o55)                             21 n + 19
(%i56) expand(x*(x+1));
                                  2
(%o56)                       441 n  + 819 n + 380
(%i57) mod(380,21);
(%o57)                                 2
(%i58) x:21*n+20;
(%o58)                             21 n + 20
(%i61) expand(x*(x+1));
                                  2
(%o61)                       441 n  + 861 n + 420
(%i62) mod(420,21);
(%o62)                                 0
(%i63)

mod(41,21)=20より、mod(x(x+1),21)=1でなければならない。しかしmod(x(x+1),21)=0,2,5,6,9,12,14,20より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時26分34秒
返信・引用
  (%i1) solve(p^2+p+41=k*17,p);
                   sqrt(68 k - 163) + 1      sqrt(68 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i2) solve(68*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        17
において、
(%i3) x:17*n;
(%o3)                                17 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            289 n  + 17 n
(%i5) x:17*n+1;
(%o5)                              17 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          289 n  + 51 n + 2
(%i7) x:17*n+2;
(%o7)                              17 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                          289 n  + 85 n + 6
(%i9) x:17*n+3;
(%o9)                              17 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                        289 n  + 119 n + 12
(%i11) x:17*n+4;
(%o11)                             17 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                        289 n  + 153 n + 20
(%i13) mod(20,17);
(%o13)                                 3
(%i14) x:17*n+5;
(%o14)                             17 n + 5
(%i15) expand(x*(x+1));
                                   2
(%o15)                        289 n  + 187 n + 30
(%i16) mod(30,17);
(%o16)                                13
(%i17) x:17*n+6;
(%o17)                             17 n + 6
(%i18) expand(x*(x+1));
                                   2
(%o18)                        289 n  + 221 n + 42
(%i19) mod(42,17);
(%o19)                                 8
(%i20) x:17*n+7;
(%o20)                             17 n + 7
(%i21) expand(x*(x+1));
                                   2
(%o21)                        289 n  + 255 n + 56
(%i22) mod(56,17);
(%o22)                                 5
(%i23) x:17*n+8;
(%o23)                             17 n + 8
(%i24) expand(x*(x+1));
                                   2
(%o24)                        289 n  + 289 n + 72
(%i25) mod(72,17);
(%o25)                                 4
(%i26) x:17*n+9;
(%o26)                             17 n + 9
(%i27) expand(x*(x+1));
                                   2
(%o27)                        289 n  + 323 n + 90
(%i28) mod(90,17);
(%o28)                                 5
(%i29) x:17*n+10;
(%o29)                             17 n + 10
(%i30) expand(x*(x+1));
                                  2
(%o30)                       289 n  + 357 n + 110
(%i31) mod(110,17);
(%o31)                                 8
(%i32) x:17*n+11;
(%o32)                             17 n + 11
(%i33) expand(x*(x+1));
                                  2
(%o33)                       289 n  + 391 n + 132
(%i34) mod(132,17);
(%o34)                                13
(%i35) x:17*n+12;
(%o35)                             17 n + 12
(%i36) expand(x*(x+1));
                                  2
(%o36)                       289 n  + 425 n + 156
(%i37) mod(156,17);
(%o37)                                 3
(%i38) x:17*n+13;
(%o38)                             17 n + 13
(%i39) expand(x*(x+1));
                                  2
(%o39)                       289 n  + 459 n + 182
(%i40) mod(182,17);
(%o40)                                12
(%i41) x:17*n+14;
(%o41)                             17 n + 14
(%i42) expand(x*(x+1));
                                  2
(%o42)                       289 n  + 493 n + 210
(%i43) mod(210,17);
(%o43)                                 6
(%i44) x:17*n+15;
(%o44)                             17 n + 15
(%i45) expand(x*(x+1));
                                  2
(%o45)                       289 n  + 527 n + 240
(%i46) mod(240,17);
(%o46)                                 2
(%i47) x:17*n+16;
(%o47)                             17 n + 16
(%i48) expand(x*(x+1));
                                  2
(%o48)                       289 n  + 561 n + 272
(%i49) mod(272,17);
(%o49)                                 0
(%i50)
mod(41,17)=7より、mod(x(x+1),17)=10でなければならない。しかしmod(x(x+1),17)=0,2,3,4,5,6,8,12,13より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時25分53秒
返信・引用
  (%i1) solve(p^2+p+41=k*15,p);
                   sqrt(60 k - 163) + 1      sqrt(60 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i2) solve(60*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        15
において、
(%i3) x:15*n;
(%o3)                                15 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            225 n  + 15 n
(%i5) x:15*n+1;
(%o5)                              15 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          225 n  + 45 n + 2
(%i7) x:15*n+2;
(%o7)                              15 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                          225 n  + 75 n + 6
(%i9) x:15*n+3;
(%o9)                              15 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                        225 n  + 105 n + 12
(%i11) x:15*n+4;
(%o11)                             15 n + 4
(%i12) expand(x*(x+1));
                                   2
(%o12)                        225 n  + 135 n + 20
(%i13) mod(20,15);
(%o13)                                 5
(%i14) x:15*n+5;
(%o14)                             15 n + 5
(%i15) expand(x*(x+1));
                                   2
(%o15)                        225 n  + 165 n + 30
(%i16) mod(30,15);
(%o16)                                 0
(%i17) x:15*n+6;
(%o17)                             15 n + 6
(%i18) expand(x*(x+1));
                                   2
(%o18)                        225 n  + 195 n + 42
(%i19) mod(42,15);
(%o19)                                12
(%i20) x:15*n+7;
(%o20)                             15 n + 7
(%i21) expand(x*(x+1));
                                   2
(%o21)                        225 n  + 225 n + 56
(%i22) mod(56,15);
(%o22)                                11
(%i23) x:15*n+8;
(%o23)                             15 n + 8
(%i24) expand(x*(x+1));
                                   2
(%o24)                        225 n  + 255 n + 72
(%i25) mod(72,15);
(%o25)                                12
(%i27) x:15*n+9;
(%o27)                             15 n + 9
(%i28) expand(x*(x+1));
                                   2
(%o28)                        225 n  + 285 n + 90
(%i29) mod(90,15);
(%o29)                                 0
(%i30) x:15*n+10;
(%o30)                             15 n + 10
(%i31) expand(x*(x+1));
                                  2
(%o31)                       225 n  + 315 n + 110
(%i32) mod(110,15);
(%o32)                                 5
(%i33) x:15*n+11;
(%o33)                             15 n + 11
(%i34) expand(x*(x+1));
                                  2
(%o34)                       225 n  + 345 n + 132
(%i35) mod(132,15);
(%o35)                                12
(%i36) x:15*n+12;
(%o36)                             15 n + 12
(%i37) expand(x*(x+1));
                                  2
(%o37)                       225 n  + 375 n + 156
(%i38) mod(156,15);
(%o38)                                 6
(%i39) x:15*n+13;
(%o39)                             15 n + 13
(%i40) expand(x*(x+1));
                                  2
(%o40)                       225 n  + 405 n + 182
(%i41) mod(182,15);
(%o41)                                 2
(%i42) x:15*n+14;
(%o42)                             15 n + 14
(%i43) expand(x*(x+1));
                                  2
(%o43)                       225 n  + 435 n + 210
(%i44) mod(210,15);
(%o44)                                 0
mod(41,15)=11より、mod(x(x+1),15)=4でなければならない。しかしmod(x(x+1),15)=0,2,5,6,11,12より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時25分13秒
返信・引用
  (%i1) solve(p^2+p+41=k*13,p);
                   sqrt(52 k - 163) + 1      sqrt(52 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i2) solve(52*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        13
において、
(%i4) x:13*n;
(%o4)                                13 n
(%i5) expand(x*(x+1));
                                      2
(%o5)                            169 n  + 13 n
(%i6) x:13*n+1;
(%o6)                              13 n + 1
(%i7) expand(x*(x+1));
                                    2
(%o7)                          169 n  + 39 n + 2
(%i8) x:13*n+2;
(%o8)                              13 n + 2
(%i9) expand(x*(x+1));
                                    2
(%o9)                          169 n  + 65 n + 6
(%i10) x:13*n+3;
(%o10)                             13 n + 3
(%i11) expand(x*(x+1));
                                   2
(%o11)                        169 n  + 91 n + 12
(%i12) x:13*n+4;
(%o12)                             13 n + 4
(%i13) expand(x*(x+1));
                                   2
(%o13)                        169 n  + 117 n + 20
(%i14) mod(20,13);
(%o14)                                 7
(%i15) x:13*n+5;
(%o15)                             13 n + 5
(%i16) expand(x*(x+1));
                                   2
(%o16)                        169 n  + 143 n + 30
(%i17) mod(30,13);
(%o17)                                 4
(%i18) x:13*n+6;
(%o18)                             13 n + 6
(%i19) expand(x*(x+1));
                                   2
(%o19)                        169 n  + 169 n + 42
(%i20) mod(42,13);
(%o20)                                 3
(%i21) x:13*n+7;
(%o21)                             13 n + 7
(%i22) expand(x*(x+1));
                                   2
(%o22)                        169 n  + 195 n + 56
(%i23) mod(56,13);
(%o23)                                 4
(%i24) x:13*n+8;
(%o24)                             13 n + 8
(%i25) expand(x*(x+1));
                                   2
(%o25)                        169 n  + 221 n + 72
(%i26) mod(72,13);
(%o26)                                 7
(%i27) x:13*n+9;
(%o27)                             13 n + 9
(%i28) expand(x*(x+1));
                                   2
(%o28)                        169 n  + 247 n + 90
(%i29) mod(90,13);
(%o29)                                12
(%i30) x:13*n+10;
(%o30)                             13 n + 10
(%i31) expand(x*(x+1));
                                  2
(%o31)                       169 n  + 273 n + 110
(%i32) mod(110,13);
(%o32)                                 6
(%i33) x:13*n+11;
(%o33)                             13 n + 11
(%i34) expand(x*(x+1));
                                  2
(%o34)                       169 n  + 299 n + 132
(%i35) mod(132,13);
(%o35)                                 2
(%i36) x:13*n+12;
(%o36)                             13 n + 12
(%i37) expand(x*(x+1));
                                  2
(%o37)                       169 n  + 325 n + 156
(%i38) mod(156,13);
(%o38)                                 0
mod(41,13)=2より、mod(x(x+1),13)=11でなければならない。しかしmod(x(x+1),11)=0,2,3,4,6,7,12より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時24分23秒
返信・引用
  (%i1) solve(p^2+p+41=k*11,p);
                   sqrt(44 k - 163) + 1      sqrt(44 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i2) solve(44*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o2)                          [k = -----------]
                                        11
において、
(%i3) x:11*n;
(%o3)                                11 n
(%i4) expand(x*(x+1));
                                      2
(%o4)                            121 n  + 11 n
(%i5) x:11*n+1;
(%o5)                              11 n + 1
(%i6) expand(x*(x+1));
                                    2
(%o6)                          121 n  + 33 n + 2
(%i7) x:11*n+2;
(%o7)                              11 n + 2
(%i8) expand(x*(x+1));
                                    2
(%o8)                          121 n  + 55 n + 6
(%i9) x:11*n+3;
(%o9)                              11 n + 3
(%i10) expand(x*(x+1));
                                   2
(%o10)                        121 n  + 77 n + 12
(%i11) mod(12,11);
(%o11)                                 1
(%i12) x:11*n+4;
(%o12)                             11 n + 4
(%i13) expand(x*(x+1));
                                   2
(%o13)                        121 n  + 99 n + 20
(%i14) mod(20,11);
(%o14)                                 9
(%i15) x:11*n+5;
(%o15)                             11 n + 5
(%i16) expand(x*(x+1));
                                   2
(%o16)                        121 n  + 121 n + 30
(%i17) mod(30,11);
(%o17)                                 8
(%i18) x:11*n+6;
(%o18)                             11 n + 6
(%i19) expand(x*(x+1));
                                   2
(%o19)                        121 n  + 143 n + 42
(%i20) mod(42,11);
(%o20)                                 9
(%i21) x:11*n+7;
(%o21)                             11 n + 7
(%i22) expand(x*(x+1));
                                   2
(%o22)                        121 n  + 165 n + 56
(%i23) mod(56,11);
(%o23)                                 1
(%i24) x:11*n+8;
(%o24)                             11 n + 8
(%i25) expand(x*(x+1));
                                   2
(%o25)                        121 n  + 187 n + 72
(%i26) mod(72,11);
(%o26)                                 6
(%i27) x:11*n+9;
(%o27)                             11 n + 9
(%i28) expand(x*(x+1));
                                   2
(%o28)                        121 n  + 209 n + 90
(%i29) mod(90,11);
(%o29)                                 2
(%i30) x:11*n+10;
(%o30)                             11 n + 10
(%i31) expand(x*(x+1));
                                  2
(%o31)                       121 n  + 231 n + 110
(%i32) mod(110,11);
(%o32)                                 0

mod(41,11)=8より、mod(x(x+1),11)=3でなければならない。しかしmod(x(x+1),11)=0,1,2,6,8,9より、kは整数にならない。
 

Re: オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時23分35秒
返信・引用
  (%i1) solve(p^2+p+41=k*y,p);
                  sqrt(4 k y - 163) + 1      sqrt(4 k y - 163) - 1
(%o1)      [p = - ---------------------, p = ---------------------]
                            2                          2
y>41ならば、mod(41,y)=41である。なぜなら41は素数だから、yと共通の因数はない。
そこで、y=41+2aとする。なぜならyは奇数だから。 sqrt(4 k y - 163) - 1=2pより、sqrt(4 k y - 163)は奇数である。なので、4ky-163=(2x+1)^2である。
(%i4) y:41+2*a;
(%o4)                              2 a + 41
(%i5) solve(4*k*y-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o5)                          [k = -----------]
                                     2 a + 41
x^2+xをyで割ったあまりは、41をyで割ったあまりを加えると41でなければならないので、
mod(x^2+x,y)=y-41=41+2a-41=2a
よって、
(%i2) y:41+2*a;
(%o2)                              2 a + 41
y>x^2+x≧0のとき、
(%i7) solve(x*(x+1)=2*a,x);
                      sqrt(8 a + 1) + 1      sqrt(8 a + 1) - 1
(%o7)          [x = - -----------------, x = -----------------]
                              2                      2
a=1のとき、x=1ゆえに、k=1,p=2ところがk>pより、NG.
a=3のとき、x=2ゆえに、k=1,p=4ところがk>pより、NG.
a=6のとき、x=3ゆえに、k=1,p=6ところがk>pより、NG.
y>x^2+x>0の場合k=1なので、ところがk>pより、NG.

2y>x^2+x≧yのとき、
(%i3) solve(x*(x+1)-y=2*a,x);
                   sqrt(16 a + 165) + 1      sqrt(16 a + 165) - 1
(%o3)       [x = - --------------------, x = --------------------]
                            2                         2
そこで、
(%i1) solve(2*x=sqrt(16*a+165)-1,a);
Is 2 x + 1 positive, negative or zero? p;
                                     2
                                    x  + x - 41
(%o1)                          [a = -----------]
                                         4
xが偶数でも奇数でも分子は奇数なので、aは自然数でない。a>0であるからy=41+2aなのでyは自然数でない。
ゆえに、p^2+p+41=kyは成立しない。

3y>x^2+x≧2yのとき、
(%i4) solve(x*(x+1)-2*y=2*a,x);
                   sqrt(24 a + 329) + 1      sqrt(24 a + 329) - 1
(%o4)       [x = - --------------------, x = --------------------]
                            2                         2
そこで、
(%i1) solve(2*x=sqrt(16*a+329)-1,a);
Is 2 x + 1 positive, negative or zero? p;
                                     2
                                    x  + x - 82
(%o1)                          [a = -----------]
                                         4

x=9のとき、a=2でy=45 3y>x^2+x≧2yであるから135>90=90 k=131/45でNG
x=10のとき、a=7でy=55 3y>x^2+x≧2yであるから165>110=110 k=151/55
x=13のとき、a=25でy=91 3y>x^2+x≧2yであるから273>182=182



4y>x^2+x>3yのとき、
(%i5) solve(x*(x+1)-3*y=2*a,x);
                   sqrt(32 a + 493) + 1      sqrt(32 a + 493) - 1
(%o5)       [x = - --------------------, x = --------------------]
                            2                         2
より、mod(x*(x+1),y)=2aなので、
(m+1)y>x^2+x>myのとき、
(%i6) solve(x*(x+1)-m*y=2*a,x);
             sqrt((8 a + 164) m + 8 a + 1) + 1
(%o6) [x = - ---------------------------------,
                             2
                                             sqrt((8 a + 164) m + 8 a + 1) - 1
                                         x = ---------------------------------]
                                                             2
 

オイラーの素数生成式

 投稿者:うんざりはちべえ  投稿日:2018年 5月 5日(土)18時22分32秒
返信・引用
  (%i1) solve(p^2+p+41=k*9,p);
                   sqrt(36 k - 163) + 1      sqrt(36 k - 163) - 1
(%o1)       [p = - --------------------, p = --------------------]
                            2                         2
より、
(%i3) solve(36*k-163=(2*x+1)^2,k);
                                     2
                                    x  + x + 41
(%o3)                          [k = -----------]
                                         9
において、
(%i4) x:9*n;
(%o4)                                 9 n
(%i5) expand(x*(x+1));
                                      2
(%o5)                             81 n  + 9 n
(%i6) x:9*n+1;
(%o6)                               9 n + 1
(%i7) expand(x*(x+1));
                                   2
(%o7)                          81 n  + 27 n + 2
(%i8) x:9*n+2;
(%o8)                               9 n + 2
(%i9) expand(x*(x+1));
                                   2
(%o9)                          81 n  + 45 n + 6
(%i10) x:9*n+3;
(%o10)                              9 n + 3
(%i11) expand(x*(x+1));
                                   2
(%o11)                         81 n  + 63 n + 12
(%i22) mod(12,9);
(%o22)                                 3
(%i12) x:9*n+4;
(%o12)                              9 n + 4
(%i13) expand(x*(x+1));
                                   2
(%o13)                         81 n  + 81 n + 20
(%i23) mod(20,9);
(%o23)                                 2
(%i14) x:9*n+5;
(%o14)                              9 n + 5
(%i15) expand(x*(x+1));
                                   2
(%o15)                         81 n  + 99 n + 30
(%i24) mod(30,9);
(%o24)                                 3
(%i16) x:9*n+6;
(%o16)                              9 n + 6
(%i17) expand(x*(x+1));
                                  2
(%o17)                        81 n  + 117 n + 42
(%i25) mod(42,9);
(%o25)                                 6
(%i18) x:9*n+7;
(%o18)                              9 n + 7
(%i19) expand(x*(x+1));
                                  2
(%o19)                        81 n  + 135 n + 56
(%i26) mod(56,9);
(%o26)                                 2
(%i20) x:9*n+8;
(%o20)                              9 n + 8
(%i21) expand(x*(x+1));
                                  2
(%o21)                        81 n  + 153 n + 72
(%i27) mod(72,9);
(%o27)                                 0

mod(41,9)=5より、mod(x(x+1),9)=4でなければならない。しかしmod(x(x+1),9)=0,2,3,6より、kは整数にならない。
 

Re: フェルマーの最終定理

 投稿者:うんざりはちべえ  投稿日:2018年 4月 8日(日)12時37分9秒
返信・引用
  x^n+y^n=z^n
x^(n-1) x  +y^(n-1) y  =z^(n-1) z   ①
x^(n-2) x^2+y^(n-2) y^2=z^(n-2) z^2 ②
x^(n-3) x^3+y^(n-3) y^3=z^(n-3) z^3 ③
x^(n-4) x^4+y^(n-4) y^4=z^(n-4) z^4 ④
         ・
         ・
         ・
x^2 x^(n-2)+y^2 y^(n-2)=z^2 z^(n-2)
x x^(n-1)  +y y^(n-1)  =z   z^(n-1)
x^n+y^n=z^n
であるから、n=2mなら、
x^(2m-m) x^m+y^(2m-m) y^m=z^(2m-m) z^m
x^m x^m+y^m y^m=z^m z^m
で折り返す。

ところで、②+①とか②ー①とかしたらどうだろう?
 

Re: フェルマーの最終定理

 投稿者:うんざりはちべえ  投稿日:2018年 4月 8日(日)12時26分13秒
返信・引用
  ソフィー・ジェルマンは、
https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9E%E3%83%BC%E3%81%AE%E6%9C%80%E7%B5%82%E5%AE%9A%E7%90%86
にどうやって証明したかが書かれています。
 

Re: フェルマーの最終定理

 投稿者:うんざりはちべえ  投稿日:2018年 4月 7日(土)16時53分34秒
返信・引用
  x^n+y^n=z^n
ならば、
x^(n-1) x+ y^(n-1) y= z^(n-1) z
が成り立つ。
ゆえに、
x^(6m+1)+y^(6m+1)=z^(6m+1)
x^6m x+ y^6m y= z^6m z
である。
 

Re: フェルマーの最終定理

 投稿者:うんざりはちべえ  投稿日:2018年 4月 7日(土)14時07分34秒
返信・引用
  x^(6m+1)+y^(6m+1)=z^(6m-1)
x^6m x+y^6m y=z^6m /z
(x^m)^6 x +(y^m)^6 y= (z^m)^6 /z
{(x^m)^2}^3 x +{(y^m)^2}^3 y= {(z^m)^2}^3 /z
{(x^m)^2}^3 xz +{(y^m)^2}^3 yz= {(z^m)^2}^3
だから、
x^(6m-1)+y^(6m+1)=z^(6m+1)
x^(6m+1)+y^(6m-1)=z^(6m+1)
も同様
x^(6m-1)+y^(6m-1)=z^(6m+1)
なら、xyをかけて、
{(x^m)^2}^3 y +{(y^m)^2}^3 x= {(z^m)^2}^3 xyz

x^(6m-1)+y^(6m-1)=z^(6m-1)
なら、xyzをかけて、
{(x^m)^2}^3 yz +{(y^m)^2}^3 xz= {(z^m)^2}^3 xy

 

Re: フェルマーの最終定理

 投稿者:うんざりはちべえ  投稿日:2018年 4月 7日(土)14時01分22秒
返信・引用
  > すると、n=6m±1で考えれば、2とおり。

x^(6m+1)+y^(6m+1)=z^(6m+1)
x^6m x+y^6m y=z^6m z
(x^m)^6 x +(y^m)^6 y= (z^m)^6 z
{(x^m)^2}^3 x +{(y^m)^2}^3 y= {(z^m)^2}^3 z

x,y,zはどうやって、消したか・・・
 

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