Pythagorean triangles – each side is a sum of two squares

 
 

Find Pythagorean triangles each of whose sides is a sum of two squares.

For example,     (9, \; 40, \; 41)

9 \; = \; 3^2
40 \; = \; 2^2 \; + \; 6^2
41 \; = \; 4^2 \; + \; 5^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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One Response to Pythagorean triangles – each side is a sum of two squares

  1. Paul says:

    here are those with a <= 1000, Format is
    {a, b, c}, a = {x^2, y^2} etc

    {45,200,205} , 45 = {3,6} , 200 = {2,14} , 205 = {3,14}
    {73,2664,2665} , 73 = {3,8} , 2664 = {30,42} , 2665 = {8,51}
    {85,720,725} , 85 = {2,9} , 720 = {12,24} , 725 = {7,26}
    {90,400,410} , 90 = {3,9} , 400 = {12,16} , 410 = {7,19}
    {104,153,185} , 104 = {2,10} , 153 = {3,12} , 185 = {4,13}
    {117,520,533} , 117 = {6,9} , 520 = {6,22} , 533 = {2,23}
    {145,10512,10513} , 145 = {1,12} , 10512 = {36,96} , 10513 = {72,73}
    {146,5328,5330} , 146 = {5,11} , 5328 = {12,72} , 5330 = {1,73}
    {153,680,697} , 153 = {3,12} , 680 = {2,26} , 697 = {11,24}
    {170,1440,1450} , 170 = {1,13} , 1440 = {12,36} , 1450 = {9,37}
    {180,800,820} , 180 = {6,12} , 800 = {4,28} , 820 = {6,28}
    {208,306,370} , 208 = {8,12} , 306 = {9,15} , 370 = {3,19}
    {221,1872,1885} , 221 = {5,14} , 1872 = {24,36} , 1885 = {6,43}
    {225,272,353} , 225 = {9,12} , 272 = {4,16} , 353 = {8,17}
    {225,1000,1025} , 225 = {9,12} , 1000 = {10,30} , 1025 = {1,32}
    {233,27144,27145} , 233 = {8,13} , 27144 = {30,162} , 27145 = {24,163}
    {234,1040,1066} , 234 = {3,15} , 1040 = {4,32} , 1066 = {15,29}
    {261,1160,1189} , 261 = {6,15} , 1160 = {2,34} , 1189 = {10,33}
    {289,2448,2465} , 289 = {8,15} , 2448 = {12,48} , 2465 = {8,49}
    {289,41760,41761} , 289 = {8,15} , 41760 = {12,204} , 41761 = {144,145}
    {290,21024,21026} , 290 = {1,17} , 21024 = {60,132} , 21026 = {1,145}
    {292,10656,10660} , 292 = {6,16} , 10656 = {60,84} , 10660 = {16,102}
    {306,1360,1394} , 306 = {9,15} , 1360 = {8,36} , 1394 = {5,37}
    {325,360,485} , 325 = {1,18} , 360 = {6,18} , 485 = {1,22}
    {328,1665,1697} , 328 = {2,18} , 1665 = {12,39} , 1697 = {4,41}
    {333,1480,1517} , 333 = {3,18} , 1480 = {6,38} , 1517 = {19,34}
    {340,2880,2900} , 340 = {4,18} , 2880 = {24,48} , 2900 = {14,52}
    {360,1600,1640} , 360 = {6,18} , 1600 = {24,32} , 1640 = {14,38}
    {360,2009,2041} , 360 = {6,18} , 2009 = {28,35} , 2041 = {4,45}
    {365,13320,13325} , 365 = {2,19} , 13320 = {18,114} , 13325 = {10,115}
    {369,800,881} , 369 = {12,15} , 800 = {4,28} , 881 = {16,25}
    {369,1640,1681} , 369 = {12,15} , 1640 = {14,38} , 1681 = {9,40}
    {405,1800,1845} , 405 = {9,18} , 1800 = {6,42} , 1845 = {9,42}
    {405,16400,16405} , 405 = {9,18} , 16400 = {4,128} , 16405 = {23,126}
    {416,612,740} , 416 = {4,20} , 612 = {6,24} , 740 = {8,26}
    {425,3600,3625} , 425 = {5,20} , 3600 = {36,48} , 3625 = {5,60}
    {442,3744,3770} , 442 = {1,21} , 3744 = {12,60} , 3770 = {7,61}
    {450,544,706} , 450 = {3,21} , 544 = {12,20} , 706 = {9,25}
    {450,2000,2050} , 450 = {3,21} , 2000 = {8,44} , 2050 = {5,45}
    {466,54288,54290} , 466 = {5,21} , 54288 = {48,228} , 54290 = {1,233}
    {468,2080,2132} , 468 = {12,18} , 2080 = {12,44} , 2132 = {4,46}
    {477,2120,2173} , 477 = {6,21} , 2120 = {2,46} , 2173 = {18,43}
    {493,4176,4205} , 493 = {3,22} , 4176 = {24,60} , 4205 = {19,62}
    {520,765,925} , 520 = {6,22} , 765 = {6,27} , 925 = {5,30}
    {521,135720,135721} , 521 = {11,20} , 135720 = {42,366} , 135721 = {260,261}
    {522,2320,2378} , 522 = {9,21} , 2320 = {4,48} , 2378 = {13,47}
    {549,2440,2501} , 549 = {15,18} , 2440 = {18,46} , 2501 = {1,50}
    {577,166464,166465} , 577 = {1,24} , 166464 = {192,360} , 166465 = {1,408}
    {578,4896,4930} , 578 = {7,23} , 4896 = {36,60} , 4930 = {13,69}
    {578,83520,83522} , 578 = {7,23} , 83520 = {24,288} , 83522 = {1,289}
    {580,42048,42052} , 580 = {2,24} , 42048 = {72,192} , 42052 = {144,146}
    {584,21312,21320} , 584 = {10,22} , 21312 = {24,144} , 21320 = {2,146}
    {585,928,1097} , 585 = {3,24} , 928 = {12,28} , 1097 = {16,29}
    {585,2600,2665} , 585 = {3,24} , 2600 = {10,50} , 2665 = {8,51}
    {585,171112,171113} , 585 = {3,24} , 171112 = {126,394} , 171113 = {37,412}
    {592,1305,1433} , 592 = {4,24} , 1305 = {3,36} , 1433 = {8,37}
    {612,2720,2788} , 612 = {6,24} , 2720 = {4,52} , 2788 = {22,48}
    {629,5328,5365} , 629 = {2,25} , 5328 = {12,72} , 5365 = {6,73}
    {650,720,970} , 650 = {5,25} , 720 = {12,24} , 970 = {3,31}
    {656,3330,3394} , 656 = {16,20} , 3330 = {9,57} , 3394 = {37,45}
    {657,2624,2705} , 657 = {9,24} , 2624 = {32,40} , 2705 = {1,52}
    {657,2920,2993} , 657 = {9,24} , 2920 = {2,54} , 2993 = {17,52}
    {657,23976,23985} , 657 = {9,24} , 23976 = {90,126} , 23985 = {24,153}
    {666,2960,3034} , 666 = {15,21} , 2960 = {16,52} , 3034 = {3,55}
    {680,5760,5800} , 680 = {2,26} , 5760 = {24,72} , 5800 = {18,74}
    {680,7209,7241} , 680 = {2,26} , 7209 = {45,72} , 7241 = {4,85}
    {697,5904,5945} , 697 = {11,24} , 5904 = {48,60} , 5945 = {4,77}
    {720,1961,2089} , 720 = {12,24} , 1961 = {5,44} , 2089 = {8,45}
    {720,3200,3280} , 720 = {12,24} , 3200 = {8,56} , 3280 = {12,56}
    {720,4018,4082} , 720 = {12,24} , 4018 = {7,63} , 4082 = {19,61}
    {725,52560,52565} , 725 = {7,26} , 52560 = {24,228} , 52565 = {71,218}
    {730,26640,26650} , 730 = {1,27} , 26640 = {48,156} , 26650 = {9,163}
    {738,1600,1762} , 738 = {3,27} , 1600 = {24,32} , 1762 = {9,41}
    {738,3280,3362} , 738 = {3,27} , 3280 = {12,56} , 3362 = {31,49}
    {765,3400,3485} , 765 = {6,27} , 3400 = {6,58} , 3485 = {2,59}
    {765,6480,6525} , 765 = {6,27} , 6480 = {36,72} , 6525 = {21,78}
    {801,3560,3649} , 801 = {15,24} , 3560 = {14,58} , 3649 = {7,60}
    {801,3920,4001} , 801 = {15,24} , 3920 = {28,56} , 4001 = {40,49}
    {801,320800,320801} , 801 = {15,24} , 320800 = {52,564} , 320801 = {215,524}
    {809,327240,327241} , 809 = {5,28} , 327240 = {126,558} , 327241 = {285,496}
    {810,3600,3690} , 810 = {9,27} , 3600 = {36,48} , 3690 = {21,57}
    {810,32800,32810} , 810 = {9,27} , 32800 = {20,180} , 32810 = {7,181}
    {832,1224,1480} , 832 = {16,24} , 1224 = {18,30} , 1480 = {6,38}
    {845,936,1261} , 845 = {2,29} , 936 = {6,30} , 1261 = {6,35}
    {848,2745,2873} , 848 = {8,28} , 2745 = {12,51} , 2873 = {8,53}
    {850,7200,7250} , 850 = {3,29} , 7200 = {12,84} , 7250 = {5,85}
    {873,3880,3977} , 873 = {12,27} , 3880 = {6,62} , 3977 = {16,61}
    {884,7488,7540} , 884 = {10,28} , 7488 = {48,72} , 7540 = {12,86}
    {900,1088,1412} , 900 = {18,24} , 1088 = {8,32} , 1412 = {16,34}
    {900,4000,4100} , 900 = {18,24} , 4000 = {20,60} , 4100 = {2,64}
    {901,7632,7685} , 901 = {1,30} , 7632 = {24,84} , 7685 = {17,86}
    {904,12753,12785} , 904 = {2,30} , 12753 = {33,108} , 12785 = {4,113}
    {909,4040,4141} , 909 = {3,30} , 4040 = {14,62} , 4141 = {35,54}
    {932,108576,108580} , 932 = {16,26} , 108576 = {60,324} , 108580 = {48,326}
    {936,1377,1665} , 936 = {6,30} , 1377 = {9,36} , 1665 = {12,39}
    {936,4160,4264} , 936 = {6,30} , 4160 = {8,64} , 4264 = {30,58}
    {936,13673,13705} , 936 = {6,30} , 13673 = {77,88} , 13705 = {4,117}
    {949,34632,34645} , 949 = {7,30} , 34632 = {6,186} , 34645 = {7,186}
    {954,4240,4346} , 954 = {15,27} , 4240 = {12,64} , 4346 = {11,65}
    {981,4360,4469} , 981 = {9,30} , 4360 = {2,66} , 4469 = {25,62}
    {986,8352,8410} , 986 = {5,31} , 8352 = {36,84} , 8410 = {29,87}

    Paul.

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