## Pythagorean triangle(a,b,c) such that a, c-b, c+b are cubes

Pythagorean triangle   $(a, b, c)$   such that

$a$,    $c + b$,    $c - b$   are integral cubes,   say

$a \; = \; p^3$
$c \; + \; b \; = \; m^3$
$c \; - \; b \; = \; n^3$

Then    $c^2 \; - \; b^2 \; = \; a^2$    gives    $m \,n \; = \; p^2$

Find such Pythagorean triangles

math grad - Interest: Number theory
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### One Response to Pythagorean triangle(a,b,c) such that a, c-b, c+b are cubes

1. Paul says:

Here’s quite a few

{27,364,365}, a = 3^3, c + b = 9^3, c – b = 1^3
{64,252,260}, a = 4^3, c + b = 8^3, c – b = 2^3
{125,7812,7813}, a = 5^3, c + b = 25^3, c – b = 1^3
{216,2912,2920}, a = 6^3, c + b = 18^3, c – b = 2^3
{343,58824,58825}, a = 7^3, c + b = 49^3, c – b = 1^3
{512,2016,2080}, a = 8^3, c + b = 16^3, c – b = 4^3
{512,16380,16388}, a = 8^3, c + b = 32^3, c – b = 2^3
{729,9828,9855}, a = 9^3, c + b = 27^3, c – b = 3^3
{729,265720,265721}, a = 9^3, c + b = 81^3, c – b = 1^3
{1000,62496,62504}, a = 10^3, c + b = 50^3, c – b = 2^3
{1331,885780,885781}, a = 11^3, c + b = 121^3, c – b = 1^3
{1728,2660,3172}, a = 12^3, c + b = 18^3, c – b = 8^3
{1728,6804,7020}, a = 12^3, c + b = 24^3, c – b = 6^3
{1728,23296,23360}, a = 12^3, c + b = 36^3, c – b = 4^3
{1728,186620,186628}, a = 12^3, c + b = 72^3, c – b = 2^3
{2197,2413404,2413405}, a = 13^3, c + b = 169^3, c – b = 1^3
{2744,470592,470600}, a = 14^3, c + b = 98^3, c – b = 2^3
{3375,7448,8177}, a = 15^3, c + b = 25^3, c – b = 9^3
{3375,45500,45625}, a = 15^3, c + b = 45^3, c – b = 5^3
{3375,210924,210951}, a = 15^3, c + b = 75^3, c – b = 3^3
{3375,5695312,5695313}, a = 15^3, c + b = 225^3, c – b = 1^3
{4096,16128,16640}, a = 16^3, c + b = 32^3, c – b = 8^3
{4096,131040,131104}, a = 16^3, c + b = 64^3, c – b = 4^3
{4096,1048572,1048580}, a = 16^3, c + b = 128^3, c – b = 2^3
{4913,12068784,12068785}, a = 17^3, c + b = 289^3, c – b = 1^3
{5832,78624,78840}, a = 18^3, c + b = 54^3, c – b = 6^3
{5832,2125760,2125768}, a = 18^3, c + b = 162^3, c – b = 2^3
{6859,23522940,23522941}, a = 19^3, c + b = 361^3, c – b = 1^3
{8000,31500,32500}, a = 20^3, c + b = 40^3, c – b = 10^3
{8000,62244,62756}, a = 20^3, c + b = 50^3, c – b = 8^3
{8000,499968,500032}, a = 20^3, c + b = 100^3, c – b = 4^3
{8000,3999996,4000004}, a = 20^3, c + b = 200^3, c – b = 2^3
{9261,58460,59189}, a = 21^3, c + b = 49^3, c – b = 9^3
{9261,124852,125195}, a = 21^3, c + b = 63^3, c – b = 7^3
{9261,1588248,1588275}, a = 21^3, c + b = 147^3, c – b = 3^3
{9261,42883060,42883061}, a = 21^3, c + b = 441^3, c – b = 1^3
{10648,7086240,7086248}, a = 22^3, c + b = 242^3, c – b = 2^3
{12167,74017944,74017945}, a = 23^3, c + b = 529^3, c – b = 1^3
{13824,21280,25376}, a = 24^3, c + b = 36^3, c – b = 16^3
{13824,54432,56160}, a = 24^3, c + b = 48^3, c – b = 12^3
{13824,186368,186880}, a = 24^3, c + b = 72^3, c – b = 8^3
{13824,442260,442476}, a = 24^3, c + b = 96^3, c – b = 6^3
{13824,1492960,1493024}, a = 24^3, c + b = 144^3, c – b = 4^3
{13824,11943932,11943940}, a = 24^3, c + b = 288^3, c – b = 2^3
{15625,976500,976625}, a = 25^3, c + b = 125^3, c – b = 5^3
{15625,122070312,122070313}, a = 25^3, c + b = 625^3, c – b = 1^3
{17576,19307232,19307240}, a = 26^3, c + b = 338^3, c – b = 2^3
{19683,265356,266085}, a = 27^3, c + b = 81^3, c – b = 9^3
{19683,7174440,7174467}, a = 27^3, c + b = 243^3, c – b = 3^3
{19683,193710244,193710245}, a = 27^3, c + b = 729^3, c – b = 1^3
{21952,86436,89180}, a = 28^3, c + b = 56^3, c – b = 14^3
{21952,470340,470852}, a = 28^3, c + b = 98^3, c – b = 8^3
{21952,3764736,3764800}, a = 28^3, c + b = 196^3, c – b = 4^3
{21952,30118140,30118148}, a = 28^3, c + b = 392^3, c – b = 2^3
{24389,297411660,297411661}, a = 29^3, c + b = 841^3, c – b = 1^3
{27000,59584,65416}, a = 30^3, c + b = 50^3, c – b = 18^3
{27000,364000,365000}, a = 30^3, c + b = 90^3, c – b = 10^3
{27000,1687392,1687608}, a = 30^3, c + b = 150^3, c – b = 6^3
{27000,45562496,45562504}, a = 30^3, c + b = 450^3, c – b = 2^3
{29791,443751840,443751841}, a = 31^3, c + b = 961^3, c – b = 1^3
{32768,129024,133120}, a = 32^3, c + b = 64^3, c – b = 16^3
{32768,1048320,1048832}, a = 32^3, c + b = 128^3, c – b = 8^3
{32768,8388576,8388640}, a = 32^3, c + b = 256^3, c – b = 4^3
{32768,67108860,67108868}, a = 32^3, c + b = 512^3, c – b = 2^3
{35937,484484,485815}, a = 33^3, c + b = 99^3, c – b = 11^3
{35937,885416,886145}, a = 33^3, c + b = 121^3, c – b = 9^3
{35937,23916060,23916087}, a = 33^3, c + b = 363^3, c – b = 3^3
{35937,645733984,645733985}, a = 33^3, c + b = 1089^3, c – b = 1^3
{39304,96550272,96550280}, a = 34^3, c + b = 578^3, c – b = 2^3
{42875,51012,66637}, a = 35^3, c + b = 49^3, c – b = 25^3
{42875,2679516,2679859}, a = 35^3, c + b = 175^3, c – b = 7^3
{42875,7353000,7353125}, a = 35^3, c + b = 245^3, c – b = 5^3
{42875,919132812,919132813}, a = 35^3, c + b = 1225^3, c – b = 1^3
{46656,71820,85644}, a = 36^3, c + b = 54^3, c – b = 24^3
{46656,183708,189540}, a = 36^3, c + b = 72^3, c – b = 18^3
{46656,628992,630720}, a = 36^3, c + b = 108^3, c – b = 12^3
{46656,2125508,2126020}, a = 36^3, c + b = 162^3, c – b = 8^3
{46656,5038740,5038956}, a = 36^3, c + b = 216^3, c – b = 6^3
{46656,17006080,17006144}, a = 36^3, c + b = 324^3, c – b = 4^3
{46656,136048892,136048900}, a = 36^3, c + b = 648^3, c – b = 2^3
{50653,1282863204,1282863205}, a = 37^3, c + b = 1369^3, c – b = 1^3
{54872,188183520,188183528}, a = 38^3, c + b = 722^3, c – b = 2^3
{59319,799708,801905}, a = 39^3, c + b = 117^3, c – b = 13^3
{59319,2413040,2413769}, a = 39^3, c + b = 169^3, c – b = 9^3
{59319,65161908,65161935}, a = 39^3, c + b = 507^3, c – b = 3^3
{59319,1759371880,1759371881}, a = 39^3, c + b = 1521^3, c – b = 1^3
{64000,252000,260000}, a = 40^3, c + b = 80^3, c – b = 20^3
{64000,497952,502048}, a = 40^3, c + b = 100^3, c – b = 16^3
{64000,2047500,2048500}, a = 40^3, c + b = 160^3, c – b = 10^3
{64000,3999744,4000256}, a = 40^3, c + b = 200^3, c – b = 8^3
{64000,31999968,32000032}, a = 40^3, c + b = 400^3, c – b = 4^3
{64000,255999996,256000004}, a = 40^3, c + b = 800^3, c – b = 2^3
{68921,2375052120,2375052121}, a = 41^3, c + b = 1681^3, c – b = 1^3
{74088,467680,473512}, a = 42^3, c + b = 98^3, c – b = 18^3
{74088,998816,1001560}, a = 42^3, c + b = 126^3, c – b = 14^3
{74088,12705984,12706200}, a = 42^3, c + b = 294^3, c – b = 6^3
{74088,343064480,343064488}, a = 42^3, c + b = 882^3, c – b = 2^3
{79507,3160681524,3160681525}, a = 43^3, c + b = 1849^3, c – b = 1^3
{85184,335412,346060}, a = 44^3, c + b = 88^3, c – b = 22^3
{85184,7085988,7086500}, a = 44^3, c + b = 242^3, c – b = 8^3
{85184,56689920,56689984}, a = 44^3, c + b = 484^3, c – b = 4^3
{85184,453519612,453519620}, a = 44^3, c + b = 968^3, c – b = 2^3
{91125,201096,220779}, a = 45^3, c + b = 75^3, c – b = 27^3
{91125,257908,273533}, a = 45^3, c + b = 81^3, c – b = 25^3
{91125,1228500,1231875}, a = 45^3, c + b = 135^3, c – b = 15^3
{91125,5694948,5695677}, a = 45^3, c + b = 225^3, c – b = 9^3
{91125,33215000,33215125}, a = 45^3, c + b = 405^3, c – b = 5^3
{91125,153773424,153773451}, a = 45^3, c + b = 675^3, c – b = 3^3
{91125,4151882812,4151882813}, a = 45^3, c + b = 2025^3, c – b = 1^3
{97336,592143552,592143560}, a = 46^3, c + b = 1058^3, c – b = 2^3
{103823,5389607664,5389607665}, a = 47^3, c + b = 2209^3, c – b = 1^3
{110592,170240,203008}, a = 48^3, c + b = 72^3, c – b = 32^3
{110592,435456,449280}, a = 48^3, c + b = 96^3, c – b = 24^3
{110592,1045660,1051492}, a = 48^3, c + b = 128^3, c – b = 18^3
{110592,1490944,1495040}, a = 48^3, c + b = 144^3, c – b = 16^3
{110592,3538080,3539808}, a = 48^3, c + b = 192^3, c – b = 12^3
{110592,11943680,11944192}, a = 48^3, c + b = 288^3, c – b = 8^3
{110592,28311444,28311660}, a = 48^3, c + b = 384^3, c – b = 6^3
{110592,95551456,95551520}, a = 48^3, c + b = 576^3, c – b = 4^3
{110592,764411900,764411908}, a = 48^3, c + b = 1152^3, c – b = 2^3
{117649,20176632,20176975}, a = 49^3, c + b = 343^3, c – b = 7^3
{117649,6920643600,6920643601}, a = 49^3, c + b = 2401^3, c – b = 1^3
{125000,7812000,7813000}, a = 50^3, c + b = 250^3, c – b = 10^3
{125000,976562496,976562504}, a = 50^3, c + b = 1250^3, c – b = 2^3

Paul.