To make {(a^2 + b^2), (x^2 + y^2), (abxy)} all square

 
 
Find four integers   a, \; b, \; x, \; y   such that
a^2 \; + \; b^2
x^2 \; + \; y^2
a\cdot b\cdot x\cdot y

are all square numbers

We could solve this by finding Primitive Pythagorean triangles (PPT) such that
a\cdot b\cdot x\cdot y   is a square number

 
For example,
 

PPT:   (15, 112, 113)   and   (12, 35, 37)
112^2 \; + \; 15^2 \; = \; 12769 \; = \; 113^2
35^2 \; + \; 12^2 \; = \; 1369 \; = \; 37^2
112\times 15\times 35\times 12 \; = \; 705600 \; = \; 840^2

PPT:   (27, 364, 365)   and   (13, 84, 85)
364^2 \; + \; 27^2 \; = \; 365^2
84^2 \; + \; 13^2 \; = \; 85^2
364\times 27\times 84\times 13 \; = \; 10732176 \; = \; 3276^2

 
 
Find other solutions
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

3 Responses to To make {(a^2 + b^2), (x^2 + y^2), (abxy)} all square

  1. paul says:

    Here are a few more

    11^2 + 60^2 = 71^2
    48^2 + 55^2 = 103^2
    11 x 60 x 48 x 55 = 1320^2

    11^2 + 60^2 = 71^2
    48^2 + 55^2 = 103^2
    11 x 60 x 48 x 55 = 1320^2

    12^2 + 35^2 = 47^2
    15^2 + 112^2 = 127^2
    12 x 35 x 15 x 112 = 840^2

    12^2 + 35^2 = 47^2
    20^2 + 21^2 = 41^2
    12 x 35 x 20 x 21 = 420^2

    12^2 + 35^2 = 47^2
    20^2 + 21^2 = 41^2
    12 x 35 x 20 x 21 = 420^2

    12^2 + 35^2 = 47^2
    15^2 + 112^2 = 127^2
    12 x 35 x 15 x 112 = 840^2

    13^2 + 84^2 = 97^2
    27^2 + 364^2 = 391^2
    13 x 84 x 27 x 364 = 3276^2

    15^2 + 112^2 = 127^2
    20^2 + 21^2 = 41^2
    15 x 112 x 20 x 21 = 840^2

    15^2 + 112^2 = 127^2
    20^2 + 21^2 = 41^2
    15 x 112 x 20 x 21 = 840^2

    15^2 + 112^2 = 127^2
    12^2 + 35^2 = 47^2
    15 x 112 x 12 x 35 = 840^2

    20^2 + 21^2 = 41^2
    12^2 + 35^2 = 47^2
    20 x 21 x 12 x 35 = 420^2

    20^2 + 21^2 = 41^2
    15^2 + 112^2 = 127^2
    20 x 21 x 15 x 112 = 840^2

    20^2 + 21^2 = 41^2
    12^2 + 35^2 = 47^2
    20 x 21 x 12 x 35 = 420^2

    20^2 + 21^2 = 41^2
    15^2 + 112^2 = 127^2
    20 x 21 x 15 x 112 = 840^2

    27^2 + 364^2 = 391^2
    13^2 + 84^2 = 97^2
    27 x 364 x 13 x 84 = 3276^2

    28^2 + 195^2 = 223^2
    60^2 + 91^2 = 151^2
    28 x 195 x 60 x 91 = 5460^2

    28^2 + 195^2 = 223^2
    60^2 + 91^2 = 151^2
    28 x 195 x 60 x 91 = 5460^2

    28^2 + 195^2 = 223^2
    105^2 + 208^2 = 313^2
    28 x 195 x 105 x 208 = 10920^2

    29^2 + 420^2 = 449^2
    192^2 + 1015^2 = 1207^2
    29 x 420 x 192 x 1015 = 48720^2

    12^2 + 35^2 = 47^2
    15^2 + 112^2 = 127^2
    12 x 35 x 15 x 112 = 840^2

    40^2 + 399^2 = 439^2
    95^2 + 168^2 = 263^2
    40 x 399 x 95 x 168 = 15960^2

    40^2 + 399^2 = 439^2
    105^2 + 608^2 = 713^2
    40 x 399 x 105 x 608 = 31920^2

    40^2 + 399^2 = 439^2
    95^2 + 168^2 = 263^2
    40 x 399 x 95 x 168 = 15960^2

    48^2 + 55^2 = 103^2
    11^2 + 60^2 = 71^2
    48 x 55 x 11 x 60 = 1320^2

    48^2 + 575^2 = 623^2
    100^2 + 621^2 = 721^2
    48 x 575 x 100 x 621 = 41400^2

    51^2 + 140^2 = 191^2
    189^2 + 340^2 = 529^2
    51 x 140 x 189 x 340 = 21420^2

    52^2 + 165^2 = 217^2
    65^2 + 2112^2 = 2177^2
    52 x 165 x 65 x 2112 = 34320^2

    48^2 + 55^2 = 103^2
    11^2 + 60^2 = 71^2
    48 x 55 x 11 x 60 = 1320^2

    55^2 + 1512^2 = 1567^2
    88^2 + 105^2 = 193^2
    55 x 1512 x 88 x 105 = 27720^2

    55^2 + 1512^2 = 1567^2
    88^2 + 105^2 = 193^2
    55 x 1512 x 88 x 105 = 27720^2

    55^2 + 1512^2 = 1567^2
    160^2 + 231^2 = 391^2
    55 x 1512 x 160 x 231 = 55440^2

    56^2 + 783^2 = 839^2
    200^2 + 609^2 = 809^2
    56 x 783 x 200 x 609 = 73080^2

    57^2 + 176^2 = 233^2
    132^2 + 475^2 = 607^2
    57 x 176 x 132 x 475 = 25080^2

    60^2 + 91^2 = 151^2
    105^2 + 208^2 = 313^2
    60 x 91 x 105 x 208 = 10920^2

    60^2 + 91^2 = 151^2
    28^2 + 195^2 = 223^2
    60 x 91 x 28 x 195 = 5460^2

    65^2 + 2112^2 = 2177^2
    52^2 + 165^2 = 217^2
    65 x 2112 x 52 x 165 = 34320^2

    88^2 + 105^2 = 193^2
    160^2 + 231^2 = 391^2
    88 x 105 x 160 x 231 = 18480^2

    60^2 + 91^2 = 151^2
    105^2 + 208^2 = 313^2
    60 x 91 x 105 x 208 = 10920^2

    60^2 + 91^2 = 151^2
    28^2 + 195^2 = 223^2
    60 x 91 x 28 x 195 = 5460^2

    95^2 + 168^2 = 263^2
    105^2 + 608^2 = 713^2
    95 x 168 x 105 x 608 = 31920^2

    88^2 + 105^2 = 193^2
    160^2 + 231^2 = 391^2
    88 x 105 x 160 x 231 = 18480^2

    105^2 + 208^2 = 313^2
    28^2 + 195^2 = 223^2
    105 x 208 x 28 x 195 = 10920^2

    105^2 + 608^2 = 713^2
    95^2 + 168^2 = 263^2
    105 x 608 x 95 x 168 = 31920^2

    111^2 + 6160^2 = 6271^2
    140^2 + 1221^2 = 1361^2
    111 x 6160 x 140 x 1221 = 341880^2

    112^2 + 3135^2 = 3247^2
    165^2 + 532^2 = 697^2
    112 x 3135 x 165 x 532 = 175560^2

    119^2 + 120^2 = 239^2
    169^2 + 14280^2 = 14449^2
    119 x 120 x 169 x 14280 = 185640^2

    119^2 + 120^2 = 239^2
    169^2 + 14280^2 = 14449^2
    119 x 120 x 169 x 14280 = 185640^2

    132^2 + 475^2 = 607^2
    57^2 + 176^2 = 233^2
    132 x 475 x 57 x 176 = 25080^2

    51^2 + 140^2 = 191^2
    189^2 + 340^2 = 529^2
    51 x 140 x 189 x 340 = 21420^2

    147^2 + 10804^2 = 10951^2
    148^2 + 5475^2 = 5623^2
    147 x 10804 x 148 x 5475 = 1134420^2

    Paul

    • benvitalis says:

      There are few errors with a^2 + b^2 = c^2

      • paul says:

        Ah yes I was accessing a wrong index for the PPT, this table should be good to go. it is all those where the short leg is <=1000. feel free to delete the above post.

        11^2 + 60^2 = 61^2
        48^2 + 55^2 = 73^2
        11 x 60 x 48 x 55 = 1320^2

        11^2 + 60^2 = 61^2
        245^2 + 1188^2 = 1213^2
        11 x 60 x 245 x 1188 = 13860^2

        12^2 + 35^2 = 37^2
        15^2 + 112^2 = 113^2
        12 x 35 x 15 x 112 = 840^2

        12^2 + 35^2 = 37^2
        20^2 + 21^2 = 29^2
        12 x 35 x 20 x 21 = 420^2

        13^2 + 84^2 = 85^2
        27^2 + 364^2 = 365^2
        13 x 84 x 27 x 364 = 3276^2

        13^2 + 84^2 = 85^2
        448^2 + 975^2 = 1073^2
        13 x 84 x 448 x 975 = 21840^2

        15^2 + 112^2 = 113^2
        20^2 + 21^2 = 29^2
        15 x 112 x 20 x 21 = 840^2

        17^2 + 144^2 = 145^2
        225^2 + 272^2 = 353^2
        17 x 144 x 225 x 272 = 12240^2

        27^2 + 364^2 = 365^2
        448^2 + 975^2 = 1073^2
        27 x 364 x 448 x 975 = 65520^2

        28^2 + 195^2 = 197^2
        60^2 + 91^2 = 109^2
        28 x 195 x 60 x 91 = 5460^2

        28^2 + 195^2 = 197^2
        105^2 + 208^2 = 233^2
        28 x 195 x 105 x 208 = 10920^2

        29^2 + 420^2 = 421^2
        192^2 + 1015^2 = 1033^2
        29 x 420 x 192 x 1015 = 48720^2

        31^2 + 480^2 = 481^2
        735^2 + 30008^2 = 30017^2
        31 x 480 x 735 x 30008 = 572880^2

        33^2 + 544^2 = 545^2
        833^2 + 1056^2 = 1345^2
        33 x 544 x 833 x 1056 = 125664^2

        36^2 + 77^2 = 85^2
        252^2 + 275^2 = 373^2
        36 x 77 x 252 x 275 = 13860^2

        40^2 + 399^2 = 401^2
        95^2 + 168^2 = 193^2
        40 x 399 x 95 x 168 = 15960^2

        40^2 + 399^2 = 401^2
        105^2 + 608^2 = 617^2
        40 x 399 x 105 x 608 = 31920^2

        43^2 + 924^2 = 925^2
        704^2 + 903^2 = 1145^2
        43 x 924 x 704 x 903 = 158928^2

        45^2 + 1012^2 = 1013^2
        495^2 + 1472^2 = 1553^2
        45 x 1012 x 495 x 1472 = 182160^2

        48^2 + 55^2 = 73^2
        245^2 + 1188^2 = 1213^2
        48 x 55 x 245 x 1188 = 27720^2

        48^2 + 575^2 = 577^2
        100^2 + 621^2 = 629^2
        48 x 575 x 100 x 621 = 41400^2

        51^2 + 140^2 = 149^2
        189^2 + 340^2 = 389^2
        51 x 140 x 189 x 340 = 21420^2

        51^2 + 140^2 = 149^2
        240^2 + 14399^2 = 14401^2
        51 x 140 x 240 x 14399 = 157080^2

        52^2 + 165^2 = 173^2
        65^2 + 2112^2 = 2113^2
        52 x 165 x 65 x 2112 = 34320^2

        55^2 + 1512^2 = 1513^2
        88^2 + 105^2 = 137^2
        55 x 1512 x 88 x 105 = 27720^2

        55^2 + 1512^2 = 1513^2
        160^2 + 231^2 = 281^2
        55 x 1512 x 160 x 231 = 55440^2

        56^2 + 783^2 = 785^2
        200^2 + 609^2 = 641^2
        56 x 783 x 200 x 609 = 73080^2

        57^2 + 176^2 = 185^2
        132^2 + 475^2 = 493^2
        57 x 176 x 132 x 475 = 25080^2

        57^2 + 176^2 = 185^2
        297^2 + 304^2 = 425^2
        57 x 176 x 297 x 304 = 30096^2

        57^2 + 176^2 = 185^2
        836^2 + 1323^2 = 1565^2
        57 x 176 x 836 x 1323 = 105336^2

        60^2 + 91^2 = 109^2
        105^2 + 208^2 = 233^2
        60 x 91 x 105 x 208 = 10920^2

        65^2 + 72^2 = 97^2
        968^2 + 14625^2 = 14657^2
        65 x 72 x 968 x 14625 = 257400^2

        68^2 + 285^2 = 293^2
        931^2 + 1020^2 = 1381^2
        68 x 285 x 931 x 1020 = 135660^2

        77^2 + 2964^2 = 2965^2
        364^2 + 627^2 = 725^2
        77 x 2964 x 364 x 627 = 228228^2

        85^2 + 132^2 = 157^2
        220^2 + 459^2 = 509^2
        85 x 132 x 220 x 459 = 33660^2

        85^2 + 3612^2 = 3613^2
        688^2 + 1785^2 = 1913^2
        85 x 3612 x 688 x 1785 = 614040^2

        88^2 + 105^2 = 137^2
        160^2 + 231^2 = 281^2
        88 x 105 x 160 x 231 = 18480^2

        93^2 + 476^2 = 485^2
        336^2 + 527^2 = 625^2
        93 x 476 x 336 x 527 = 88536^2

        95^2 + 168^2 = 193^2
        105^2 + 608^2 = 617^2
        95 x 168 x 105 x 608 = 31920^2

        108^2 + 2915^2 = 2917^2
        529^2 + 139920^2 = 139921^2
        108 x 2915 x 529 x 139920 = 4827240^2

        111^2 + 6160^2 = 6161^2
        140^2 + 1221^2 = 1229^2
        111 x 6160 x 140 x 1221 = 341880^2

        111^2 + 6160^2 = 6161^2
        231^2 + 2960^2 = 2969^2
        111 x 6160 x 231 x 2960 = 683760^2

        111^2 + 6160^2 = 6161^2
        259^2 + 660^2 = 709^2
        111 x 6160 x 259 x 660 = 341880^2

        112^2 + 3135^2 = 3137^2
        165^2 + 532^2 = 557^2
        112 x 3135 x 165 x 532 = 175560^2

        119^2 + 120^2 = 169^2
        169^2 + 14280^2 = 14281^2
        119 x 120 x 169 x 14280 = 185640^2

        121^2 + 7320^2 = 7321^2
        375^2 + 7808^2 = 7817^2
        121 x 7320 x 375 x 7808 = 1610400^2

        121^2 + 7320^2 = 7321^2
        488^2 + 59535^2 = 59537^2
        121 x 7320 x 488 x 59535 = 5072760^2

        132^2 + 475^2 = 493^2
        297^2 + 304^2 = 425^2
        132 x 475 x 297 x 304 = 75240^2

        132^2 + 475^2 = 493^2
        836^2 + 1323^2 = 1565^2
        132 x 475 x 836 x 1323 = 263340^2

        132^2 + 1085^2 = 1093^2
        341^2 + 420^2 = 541^2
        132 x 1085 x 341 x 420 = 143220^2

        132^2 + 1085^2 = 1093^2
        385^2 + 1488^2 = 1537^2
        132 x 1085 x 385 x 1488 = 286440^2

        140^2 + 1221^2 = 1229^2
        231^2 + 2960^2 = 2969^2
        140 x 1221 x 231 x 2960 = 341880^2

        140^2 + 1221^2 = 1229^2
        259^2 + 660^2 = 709^2
        140 x 1221 x 259 x 660 = 170940^2

        147^2 + 10804^2 = 10805^2
        148^2 + 5475^2 = 5477^2
        147 x 10804 x 148 x 5475 = 1134420^2

        165^2 + 1508^2 = 1517^2
        580^2 + 21021^2 = 21029^2
        165 x 1508 x 580 x 21021 = 1741740^2

        189^2 + 340^2 = 389^2
        240^2 + 14399^2 = 14401^2
        189 x 340 x 240 x 14399 = 471240^2

        209^2 + 21840^2 = 21841^2
        741^2 + 1540^2 = 1709^2
        209 x 21840 x 741 x 1540 = 2282280^2

        231^2 + 2960^2 = 2969^2
        259^2 + 660^2 = 709^2
        231 x 2960 x 259 x 660 = 341880^2

        241^2 + 29040^2 = 29041^2
        972^2 + 59045^2 = 59053^2
        241 x 29040 x 972 x 59045 = 20041560^2

        252^2 + 3965^2 = 3973^2
        549^2 + 1820^2 = 1901^2
        252 x 3965 x 549 x 1820 = 999180^2

        252^2 + 3965^2 = 3973^2
        585^2 + 6832^2 = 6857^2
        252 x 3965 x 585 x 6832 = 1998360^2

        276^2 + 493^2 = 565^2
        783^2 + 306544^2 = 306545^2
        276 x 493 x 783 x 306544 = 5714856^2

        280^2 + 759^2 = 809^2
        385^2 + 552^2 = 673^2
        280 x 759 x 385 x 552 = 212520^2

        287^2 + 816^2 = 865^2
        476^2 + 1107^2 = 1205^2
        287 x 816 x 476 x 1107 = 351288^2

        287^2 + 816^2 = 865^2
        484^2 + 14637^2 = 14645^2
        287 x 816 x 484 x 14637 = 1288056^2

        287^2 + 816^2 = 865^2
        525^2 + 2788^2 = 2837^2
        287 x 816 x 525 x 2788 = 585480^2

        289^2 + 41760^2 = 41761^2
        441^2 + 1160^2 = 1241^2
        289 x 41760 x 441 x 1160 = 2484720^2

        295^2 + 1728^2 = 1753^2
        540^2 + 2891^2 = 2941^2
        295 x 1728 x 540 x 2891 = 892080^2

        297^2 + 304^2 = 425^2
        836^2 + 1323^2 = 1565^2
        297 x 304 x 836 x 1323 = 316008^2

        300^2 + 22499^2 = 22501^2
        596^2 + 22197^2 = 22205^2
        300 x 22499 x 596 x 22197 = 9449580^2

        312^2 + 1505^2 = 1537^2
        455^2 + 4128^2 = 4153^2
        312 x 1505 x 455 x 4128 = 939120^2

        312^2 + 1505^2 = 1537^2
        559^2 + 840^2 = 1009^2
        312 x 1505 x 559 x 840 = 469560^2

        312^2 + 2695^2 = 2713^2
        440^2 + 1911^2 = 1961^2
        312 x 2695 x 440 x 1911 = 840840^2

        320^2 + 25599^2 = 25601^2
        371^2 + 1380^2 = 1429^2
        320 x 25599 x 371 x 1380 = 2047920^2

        341^2 + 420^2 = 541^2
        385^2 + 1488^2 = 1537^2
        341 x 420 x 385 x 1488 = 286440^2

        375^2 + 7808^2 = 7817^2
        488^2 + 59535^2 = 59537^2
        375 x 7808 x 488 x 59535 = 9223200^2

        375^2 + 70312^2 = 70313^2
        408^2 + 2585^2 = 2617^2
        375 x 70312 x 408 x 2585 = 5273400^2

        433^2 + 93744^2 = 93745^2
        867^2 + 375844^2 = 375845^2
        433 x 93744 x 867 x 375844 = 115008264^2

        455^2 + 4128^2 = 4153^2
        559^2 + 840^2 = 1009^2
        455 x 4128 x 559 x 840 = 939120^2

        476^2 + 1107^2 = 1205^2
        484^2 + 14637^2 = 14645^2
        476 x 1107 x 484 x 14637 = 1932084^2

        476^2 + 1107^2 = 1205^2
        525^2 + 2788^2 = 2837^2
        476 x 1107 x 525 x 2788 = 878220^2

        484^2 + 14637^2 = 14645^2
        525^2 + 2788^2 = 2837^2
        484 x 14637 x 525 x 2788 = 3220140^2

        495^2 + 952^2 = 1073^2
        648^2 + 6545^2 = 6577^2
        495 x 952 x 648 x 6545 = 1413720^2

        528^2 + 1025^2 = 1153^2
        656^2 + 1617^2 = 1745^2
        528 x 1025 x 656 x 1617 = 757680^2

        528^2 + 7735^2 = 7753^2
        715^2 + 1428^2 = 1597^2
        528 x 7735 x 715 x 1428 = 2042040^2

        528^2 + 7735^2 = 7753^2
        935^2 + 17472^2 = 17497^2
        528 x 7735 x 935 x 17472 = 8168160^2

        549^2 + 1820^2 = 1901^2
        585^2 + 6832^2 = 6857^2
        549 x 1820 x 585 x 6832 = 1998360^2

        560^2 + 3111^2 = 3161^2
        732^2 + 14875^2 = 14893^2
        560 x 3111 x 732 x 14875 = 4355400^2

        592^2 + 87615^2 = 87617^2
        885^2 + 1628^2 = 1853^2
        592 x 87615 x 885 x 1628 = 8644680^2

        649^2 + 1680^2 = 1801^2
        875^2 + 7788^2 = 7837^2
        649 x 1680 x 875 x 7788 = 2725800^2

        663^2 + 1216^2 = 1385^2
        988^2 + 1275^2 = 1613^2
        663 x 1216 x 988 x 1275 = 1007760^2

        715^2 + 1428^2 = 1597^2
        935^2 + 17472^2 = 17497^2
        715 x 1428 x 935 x 17472 = 4084080^2

        897^2 + 2296^2 = 2465^2
        975^2 + 52808^2 = 52817^2
        897 x 2296 x 975 x 52808 = 10297560^2

        912^2 + 3185^2 = 3313^2
        980^2 + 60021^2 = 60029^2
        912 x 3185 x 980 x 60021 = 13071240^2

        945^2 + 17848^2 = 17873^2
        968^2 + 234255^2 = 234257^2
        945 x 17848 x 968 x 234255 = 61843320^2

        Paul.

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