如何高效产生本原勾股数组？

litong

比如3 4 5和5 12 13都是勾股数组，且互质，
那么我就想问如何高效产生这些数组呢？
比如产生边长都<=1000的勾股本原数组，
按照(a,b,c)其中a<=b<=c的方式排列！

BristSes

1. Clear["Global`*"];(*清除所有变量*)aa=Flatten[#,1]&@Table[{i,j},{i,1,1000,2},{j,1,i,2}];(*产生二元数组(都是奇数),并且第一个元素大于等于第二个*)bb=Select[aa,GCD@@#==1&];(*选择互质的*)cc={#1*#2,(#1^2-#2^2)/2,(#1^2+#2^2)/2}&@@@bb;(*产生勾股数*)dd=Sort[#]&/@cc;(*排序,从小到大*)ee=Select[dd,#[[3]]<=1000&](*最大的小于1000*)

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求解结果
{{0, 1, 1}, {3, 4, 5}, {5, 12, 13}, {8, 15, 17}, {7, 24, 25}, {20, 21,
   29}, {12, 35, 37}, {9, 40, 41}, {28, 45, 53}, {16, 63, 65}, {11,
  60, 61}, {33, 56, 65}, {48, 55, 73}, {36, 77, 85}, {20, 99,
  101}, {13, 84, 85}, {39, 80, 89}, {65, 72, 97}, {60, 91, 109}, {44,
  117, 125}, {24, 143, 145}, {15, 112, 113}, {88, 105, 137}, {52, 165,
   173}, {28, 195, 197}, {17, 144, 145}, {51, 140, 149}, {85, 132,
  157}, {119, 120, 169}, {104, 153, 185}, {84, 187, 205}, {60, 221,
  229}, {32, 255, 257}, {19, 180, 181}, {57, 176, 185}, {95, 168,
  193}, {133, 156, 205}, {140, 171, 221}, {120, 209, 241}, {96, 247,
  265}, {68, 285, 293}, {36, 323, 325}, {21, 220, 221}, {105, 208,
  233}, {160, 231, 281}, {136, 273, 305}, {76, 357, 365}, {40, 399,
  401}, {23, 264, 265}, {69, 260, 269}, {115, 252, 277}, {161, 240,
  289}, {207, 224, 305}, {204, 253, 325}, {180, 299, 349}, {152, 345,
  377}, {120, 391, 409}, {84, 437, 445}, {44, 483, 485}, {25, 312,
  313}, {75, 308, 317}, {175, 288, 337}, {225, 272, 353}, {252, 275,
  373}, {228, 325, 397}, {168, 425, 457}, {132, 475, 493}, {92, 525,
  533}, {48, 575, 577}, {27, 364, 365}, {135, 352, 377}, {189, 340,
  389}, {297, 304, 425}, {280, 351, 449}, {220, 459, 509}, {184, 513,
  545}, {100, 621, 629}, {52, 675, 677}, {29, 420, 421}, {87, 416,
  425}, {145, 408, 433}, {203, 396, 445}, {261, 380, 461}, {319, 360,
  481}, {336, 377, 505}, {308, 435, 533}, {276, 493, 565}, {240, 551,
  601}, {200, 609, 641}, {156, 667, 685}, {108, 725, 733}, {56, 783,
  785}, {31, 480, 481}, {93, 476, 485}, {155, 468, 493}, {217, 456,
  505}, {279, 440, 521}, {341, 420, 541}, {396, 403, 565}, {368, 465,
  593}, {336, 527, 625}, {300, 589, 661}, {260, 651, 701}, {216, 713,
  745}, {168, 775, 793}, {116, 837, 845}, {60, 899, 901}, {33, 544,
  545}, {165, 532, 557}, {231, 520, 569}, {429, 460, 629}, {400, 561,
  689}, {364, 627, 725}, {280, 759, 809}, {232, 825, 857}, {124, 957,
  965}, {35, 612, 613}, {105, 608, 617}, {315, 572, 653}, {385, 552,
  673}, {455, 528, 697}, {468, 595, 757}, {432, 665, 793}, {348, 805,
  877}, {248, 945, 977}, {37, 684, 685}, {111, 680, 689}, {185, 672,
  697}, {259, 660, 709}, {333, 644, 725}, {407, 624, 745}, {481, 600,
  769}, {555, 572, 797}, {540, 629, 829}, {504, 703, 865}, {464, 777,
  905}, {420, 851, 949}, {372, 925, 997}, {39, 760, 761}, {195, 748,
  773}, {273, 736, 785}, {429, 700, 821}, {616, 663, 905}, {580, 741,
  941}, {41, 840, 841}, {123, 836, 845}, {205, 828, 853}, {287, 816,
  865}, {369, 800, 881}, {451, 780, 901}, {533, 756, 925}, {615, 728,
  953}, {696, 697, 985}, {43, 924, 925}, {129, 920, 929}, {215, 912,
  937}, {301, 900, 949}, {387, 884, 965}, {473, 864, 985}}

一共159组

oabipalehov

修正后的代码

1. Clear["Global`*"];(*清除所有变量*)aa=Flatten[#,1]&@Table[{i,j},{i,1,1000,2},{j,1,i,2}];(*产生二元数组(都是奇数),并且第一个元素大于等于第二个*)bb=Select[aa,GCD@@#==1&&(#[[1]]>#[[2]])&];(*选择互质并且前者大的*)cc={#1*#2,(#1^2-#2^2)/2,(#1^2+#2^2)/2}&@@@bb;(*产生勾股数*)dd=Sort[#]&/@cc;(*排序,从小到大*)ee=Select[dd,#[[3]]<=1000&](*最大的小于1000*)

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输出结果

1. {{3,4,5},{5,12,13},{8,15,17},{7,24,25},{20,21,29},{12,35,37},{9,40,41},{28,45,53},{16,63,65},{11,60,61},{33,56,65},{48,55,73},{36,77,85},{20,99,101},{13,84,85},{39,80,89},{65,72,97},{60,91,109},{44,117,125},{24,143,145},{15,112,113},{88,105,137},{52,165,173},{28,195,197},{17,144,145},{51,140,149},{85,132,157},{119,120,169},{104,153,185},{84,187,205},{60,221,229},{32,255,257},{19,180,181},{57,176,185},{95,168,193},{133,156,205},{140,171,221},{120,209,241},{96,247,265},{68,285,293},{36,323,325},{21,220,221},{105,208,233},{160,231,281},{136,273,305},{76,357,365},{40,399,401},{23,264,265},{69,260,269},{115,252,277},{161,240,289},{207,224,305},{204,253,325},{180,299,349},{152,345,377},{120,391,409},{84,437,445},{44,483,485},{25,312,313},{75,308,317},{175,288,337},{225,272,353},{252,275,373},{228,325,397},{168,425,457},{132,475,493},{92,525,533},{48,575,577},{27,364,365},{135,352,377},{189,340,389},{297,304,425},{280,351,449},{220,459,509},{184,513,545},{100,621,629},{52,675,677},{29,420,421},{87,416,425},{145,408,433},{203,396,445},{261,380,461},{319,360,481},{336,377,505},{308,435,533},{276,493,565},{240,551,601},{200,609,641},{156,667,685},{108,725,733},{56,783,785},{31,480,481},{93,476,485},{155,468,493},{217,456,505},{279,440,521},{341,420,541},{396,403,565},{368,465,593},{336,527,625},{300,589,661},{260,651,701},{216,713,745},{168,775,793},{116,837,845},{60,899,901},{33,544,545},{165,532,557},{231,520,569},{429,460,629},{400,561,689},{364,627,725},{280,759,809},{232,825,857},{124,957,965},{35,612,613},{105,608,617},{315,572,653},{385,552,673},{455,528,697},{468,595,757},{432,665,793},{348,805,877},{248,945,977},{37,684,685},{111,680,689},{185,672,697},{259,660,709},{333,644,725},{407,624,745},{481,600,769},{555,572,797},{540,629,829},{504,703,865},{464,777,905},{420,851,949},{372,925,997},{39,760,761},{195,748,773},{273,736,785},{429,700,821},{616,663,905},{580,741,941},{41,840,841},{123,836,845},{205,828,853},{287,816,865},{369,800,881},{451,780,901},{533,756,925},{615,728,953},{696,697,985},{43,924,925},{129,920,929},{215,912,937},{301,900,949},{387,884,965},{473,864,985}}

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iniogateln

nyy 发表于 2023-12-7 11:25
修正后的代码

1. Clear["Global`*"];(*清除所有变量*)max=1000;(*边长的最大值*)aa1=Flatten[#,1]&@Table[{i,j},{i,1,max,2},{j,2,i,2}];(*一个奇数,一个偶数,前者大*)aa2=Flatten[#,1]&@Table[{i,j},{i,2,max,2},{j,1,i,2}];(*一个偶数,一个奇数,前者大*)aa=Join[aa1,aa2];(*数据合并*)bb=Select[aa,GCD@@#==1&];(*选择互质*)cc={2*#1*#2,(#1^2-#2^2),(#1^2+#2^2)}&@@@bb;(*产生勾股数*)dd=Sort[#]&/@cc;(*排序,从小到大*)ee=Select[dd,#[[3]]<=max&](*最大的小于1000*)

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勾股数组

输出结果

1. {{5, 12, 13}, {20, 21, 29}, {9, 40, 41}, {28, 45, 53}, {33, 56,   65}, {13, 84, 85}, {36, 77, 85}, {65, 72, 97}, {17, 144, 145}, {44,   117, 125}, {88, 105, 137}, {85, 132, 157}, {57, 176, 185}, {21, 220,   221}, {52, 165, 173}, {104, 153, 185}, {133, 156, 205}, {105, 208,   233}, {69, 260, 269}, {25, 312, 313}, {60, 221, 229}, {120, 209,   241}, {161, 240, 289}, {29, 420, 421}, {68, 285, 293}, {136, 273,   305}, {204, 253, 325}, {225, 272, 353}, {189, 340, 389}, {145, 408,   433}, {93, 476, 485}, {33, 544, 545}, {76, 357, 365}, {152, 345,   377}, {228, 325, 397}, {297, 304, 425}, {261, 380, 461}, {217, 456,   505}, {165, 532, 557}, {105, 608, 617}, {37, 684, 685}, {84, 437,   445}, {168, 425, 457}, {336, 377, 505}, {341, 420, 541}, {185, 672,   697}, {41, 840, 841}, {92, 525, 533}, {184, 513, 545}, {276, 493,   565}, {368, 465, 593}, {429, 460, 629}, {385, 552, 673}, {333, 644,   725}, {273, 736, 785}, {205, 828, 853}, {129, 920, 929}, {100, 621,   629}, {200, 609, 641}, {300, 589, 661}, {400, 561, 689}, {481, 600,   769}, {429, 700, 821}, {369, 800, 881}, {301, 900, 949}, {108, 725,   733}, {216, 713, 745}, {432, 665, 793}, {540, 629, 829}, {533, 756,   925}, {473, 864, 985}, {116, 837, 845}, {232, 825, 857}, {348, 805,   877}, {464, 777, 905}, {580, 741, 941}, {696, 697, 985}, {124, 957,   965}, {248, 945, 977}, {372, 925, 997}, {3, 4, 5}, {8, 15, 17}, {7,   24, 25}, {12, 35, 37}, {11, 60, 61}, {16, 63, 65}, {48, 55,   73}, {39, 80, 89}, {15, 112, 113}, {20, 99, 101}, {60, 91,   109}, {51, 140, 149}, {19, 180, 181}, {24, 143, 145}, {119, 120,   169}, {95, 168, 193}, {23, 264, 265}, {28, 195, 197}, {84, 187,   205}, {140, 171, 221}, {115, 252, 277}, {75, 308, 317}, {27, 364,   365}, {32, 255, 257}, {96, 247, 265}, {160, 231, 281}, {207, 224,   305}, {175, 288, 337}, {135, 352, 377}, {87, 416, 425}, {31, 480,   481}, {36, 323, 325}, {180, 299, 349}, {252, 275, 373}, {203, 396,   445}, {155, 468, 493}, {35, 612, 613}, {40, 399, 401}, {120, 391,   409}, {280, 351, 449}, {319, 360, 481}, {279, 440, 521}, {231, 520,   569}, {111, 680, 689}, {39, 760, 761}, {44, 483, 485}, {132, 475,   493}, {220, 459, 509}, {308, 435, 533}, {396, 403, 565}, {315, 572,   653}, {259, 660, 709}, {195, 748, 773}, {123, 836, 845}, {43, 924,   925}, {48, 575, 577}, {240, 551, 601}, {336, 527, 625}, {455, 528,   697}, {407, 624, 745}, {287, 816, 865}, {215, 912, 937}, {52, 675,   677}, {156, 667, 685}, {260, 651, 701}, {364, 627, 725}, {468, 595,   757}, {555, 572, 797}, {451, 780, 901}, {387, 884, 965}, {56, 783,   785}, {168, 775, 793}, {280, 759, 809}, {504, 703, 865}, {616, 663,   905}, {615, 728, 953}, {60, 899, 901}, {420, 851, 949}}

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uahexuk

1. Clear["Global`*"];SortBy[Select[Flatten[Table[Sort@{m^2-n^2,2 m n,m^2+n^2},{n,21},{m,n+1,Sqrt[1000-n^2]}],1],GCD[#[[1]],#[[2]]]==1&],First]

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igikzari

你的代码比我快多了！

urinowiq

northwolves 发表于 2023-12-7 15:54

1. Clear["Global`*"];(*清除所有变量*)max=2000;(*边长的最大值*)nmax=1/2*(-1+Sqrt[-1+2*max]);(*由n^2+(n+1)^2<=max计算出来的n的最大值*)aa=Flatten[#,1]&@Table[Sort@{m^2-n^2,2*m*n,m^2+n^2},{n,1,nmax},{m,n+1,Sqrt[max-n^2]}];(*产生勾股数*)bb=Select[aa,GCD@@#==1&];(*选择互质*)cc=Sort[bb,#1[[3]]<#2[[3]]&](*按照第三列升序排列*)

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把你的代码肢解一下，并且添加上注释，这样更容易看明白、更好维护

1. {{3,4,5},{5,12,13},{8,15,17},{7,24,25},{20,21,29},{12,35,37},{9,40,41},{28,45,53},{11,60,61},{33,56,65},{16,63,65},{48,55,73},{13,84,85},{36,77,85},{39,80,89},{65,72,97},{20,99,101},{60,91,109},{15,112,113},{44,117,125},{88,105,137},{17,144,145},{24,143,145},{51,140,149},{85,132,157},{119,120,169},{52,165,173},{19,180,181},{57,176,185},{104,153,185},{95,168,193},{28,195,197},{133,156,205},{84,187,205},{21,220,221},{140,171,221},{60,221,229},{105,208,233},{120,209,241},{32,255,257},{23,264,265},{96,247,265},{69,260,269},{115,252,277},{160,231,281},{161,240,289},{68,285,293},{207,224,305},{136,273,305},{25,312,313},{75,308,317},{204,253,325},{36,323,325},{175,288,337},{180,299,349},{225,272,353},{27,364,365},{76,357,365},{252,275,373},{135,352,377},{152,345,377},{189,340,389},{228,325,397},{40,399,401},{120,391,409},{29,420,421},{87,416,425},{297,304,425},{145,408,433},{203,396,445},{84,437,445},{280,351,449},{168,425,457},{261,380,461},{31,480,481},{319,360,481},{93,476,485},{44,483,485},{155,468,493},{132,475,493},{217,456,505},{336,377,505},{220,459,509},{279,440,521},{308,435,533},{92,525,533},{341,420,541},{33,544,545},{184,513,545},{165,532,557},{396,403,565},{276,493,565},{231,520,569},{48,575,577},{368,465,593},{240,551,601},{35,612,613},{105,608,617},{336,527,625},{429,460,629},{100,621,629},{200,609,641},{315,572,653},{300,589,661},{385,552,673},{52,675,677},{37,684,685},{156,667,685},{111,680,689},{400,561,689},{185,672,697},{455,528,697},{260,651,701},{259,660,709},{333,644,725},{364,627,725},{108,725,733},{407,624,745},{216,713,745},{468,595,757},{39,760,761},{481,600,769},{195,748,773},{273,736,785},{56,783,785},{432,665,793},{168,775,793},{555,572,797},{280,759,809},{429,700,821},{540,629,829},{41,840,841},{123,836,845},{116,837,845},{205,828,853},{232,825,857},{287,816,865},{504,703,865},{348,805,877},{369,800,881},{451,780,901},{60,899,901},{616,663,905},{464,777,905},{43,924,925},{533,756,925},{129,920,929},{215,912,937},{580,741,941},{301,900,949},{420,851,949},{615,728,953},{387,884,965},{124,957,965},{248,945,977},{473,864,985},{696,697,985},{372,925,997},{559,840,1009},{45,1012,1013},{660,779,1021},{496,897,1025},{64,1023,1025},{192,1015,1033},{315,988,1037},{645,812,1037},{320,999,1049},{620,861,1061},{731,780,1069},{495,952,1073},{448,975,1073},{132,1085,1093},{585,928,1097},{47,1104,1105},{744,817,1105},{576,943,1105},{264,1073,1105},{141,1100,1109},{235,1092,1117},{329,1080,1129},{423,1064,1145},{704,903,1145},{528,1025,1153},{765,868,1157},{68,1155,1157},{517,1044,1165},{204,1147,1165},{340,1131,1181},{611,1020,1189},{660,989,1189},{832,855,1193},{49,1200,1201},{147,1196,1205},{476,1107,1205},{245,1188,1213},{705,992,1217},{140,1221,1229},{612,1075,1237},{441,1160,1241},{280,1209,1241},{799,960,1249},{539,1140,1261},{420,1189,1261},{748,1035,1277},{637,1116,1285},{893,924,1285},{560,1161,1289},{72,1295,1297},{51,1300,1301},{255,1288,1313},{735,1088,1313},{360,1271,1321},{357,1276,1325},{884,987,1325},{833,1056,1345},{504,1247,1345},{561,1240,1361},{840,1081,1369},{148,1365,1373},{931,1020,1381},{663,1216,1385},{296,1353,1385},{53,1404,1405},{444,1333,1405},{159,1400,1409},{265,1392,1417},{792,1175,1417},{371,1380,1429},{592,1305,1433},{477,1364,1445},{76,1443,1445},{228,1435,1453},{583,1344,1465},{936,1127,1465},{740,1269,1469},{380,1419,1469},{969,1120,1481},{689,1320,1489},{532,1395,1493},{55,1512,1513},{888,1225,1513},{165,1508,1517},{795,1292,1517},{684,1363,1525},{156,1517,1525},{385,1488,1537},{312,1505,1537},{901,1260,1549},{495,1472,1553},{1036,1173,1565},{836,1323,1565},{1007,1224,1585},{624,1457,1585},{715,1428,1597},{80,1599,1601},{240,1591,1609},{988,1275,1613},{780,1421,1621},{57,1624,1625},{1113,1184,1625},{285,1612,1637},{399,1600,1649},{560,1551,1649},{935,1368,1657},{1140,1219,1669},{720,1519,1681},{627,1564,1685},{164,1677,1685},{1045,1332,1693},{328,1665,1697},{741,1540,1709},{1092,1325,1717},{492,1645,1717},{880,1479,1721},{1155,1292,1733},{59,1740,1741},{177,1736,1745},{656,1617,1745},{295,1728,1753},{413,1716,1765},{84,1763,1765},{969,1480,1769},{1040,1431,1769},{1248,1265,1777},{531,1700,1781},{820,1581,1781},{420,1739,1789},{649,1680,1801},{767,1656,1825},{984,1537,1825},{885,1628,1853},{172,1845,1853},{61,1860,1861},{183,1856,1865},{344,1833,1865},{305,1848,1873},{1148,1485,1877},{427,1836,1885},{1003,1596,1885},{924,1643,1885},{516,1813,1885},{1311,1360,1889},{549,1820,1901},{688,1785,1913},{671,1800,1921},{1121,1560,1921},{1092,1595,1933},{1312,1425,1937},{88,1935,1937},{793,1776,1945},{264,1927,1945},{860,1749,1949},{1239,1520,1961},{440,1911,1961},{915,1748,1973},{63,1984,1985},{616,1887,1985},{1032,1705,1993},{315,1972,1997}}

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urinowiq

简单换一个思路，

1. Clear["Global`*"];(*清除所有变量*)max=1000;(*边长的最大值*)jmax=-1+Sqrt[-1+max];(*由j^2+(j+2)^2<=2*max计算出j的最大值*)aa=Flatten[#,1]&@Table[Sort@{(i*j),(i^2-j^2)/2,(i^2+j^2)/2},{j,1,jmax,2},{i,j+2,Sqrt[2*max-j^2],2}];bb=Select[aa,GCD@@#==1&];(*选择互质*)cc=Sort[bb,#1[[3]]<#2[[3]]&](*按照第三列升序排列*)

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应该比你的代码稍微快一点点

IsacSes

计算到两百万吧
上代码

1. Clear["Global`*"];(*清除所有变量*)Timing[    max=2*10^6;(*边长的最大值*)    nmax=1/2*(-1+Sqrt[-1+2*max]);(*由n^2+(n+1)^2<=max计算出来的n的最大值*)    aa=Flatten[#,1]&@Table[Sort@{m^2-n^2,2*m*n,m^2+n^2},{n,1,nmax},{m,n+1,Sqrt[max-n^2]}];(*产生勾股数*)    bb=Select[aa,GCD@@#==1&];(*选择互质*)    cc=Sort[bb,#1[[3]]<#2[[3]]&](*按照第三列升序排列*)]Clear["Global`*"];(*清除所有变量*)Timing[    max=2*10^6;(*边长的最大值*)    jmax=-1+Sqrt[-1+max];(*由j^2+(j+2)^2<=2*max计算出j的最大值*)    aa=Flatten[#,1]&@Table[Sort@{(i*j),(i^2-j^2)/2,(i^2+j^2)/2},{j,1,jmax,2},{i,j+2,Sqrt[2*max-j^2],2}];    bb=Select[aa,GCD@@#==1&];(*选择互质*)    cc=Sort[bb,#1[[3]]<#2[[3]]&](*按照第三列升序排列*)]

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输出时间
7.05125 这个是你的代码的时间
5.78764 这个是我修改后的代码的时间
从这个上来看，还是节省了一点时间的，我感觉我的代码就比你好一点点

afijozaci

这个还可以快上几倍，勾股数组的斜边的通项公式

ClearAll["Global`*"];(*清除所有变量*)
Timing[
max=2*10^6;SortBy[Sort/@Flatten[Table[{m^2-n^2,2m n,m^2+n^2},{n,(Sqrt[2max-1]-1)/2},{m,Select[Range[n+1,Sqrt[max-n^2]],GCD[#,n]==1&&OddQ@(n+#)&]}],1],Last]]