## 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

math grad - Interest: Number theory
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### 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.