## Make {x^4 + x^2 y^2, y^4 + x^2 y^2, x + y} squares

Find two positive integers   $x$   and   $y$   to make the three expressions squares

$x^4 \; + \; x^2 \, y^2$
$y^4 \; + \; x^2 \, y^2$
$x \; + \; y$

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

### One Response to Make {x^4 + x^2 y^2, y^4 + x^2 y^2, x + y} squares

1. Paul says:

Here are a few

9^4 + 9^2 X 40^2 = 369^2
40^4 + 9^2 X 40^2 = 1640^2
9 + 40 = 7^2

21^4 + 21^2 X 28^2 = 735^2
28^4 + 21^2 X 28^2 = 980^2
21 + 28 = 7^2

36^4 + 36^2 X 160^2 = 5904^2
160^4 + 36^2 X 160^2 = 26240^2
36 + 160 = 14^2

81^4 + 81^2 X 360^2 = 29889^2
360^4 + 81^2 X 360^2 = 132840^2
81 + 360 = 21^2

84^4 + 84^2 X 112^2 = 11760^2
112^4 + 84^2 X 112^2 = 15680^2
84 + 112 = 14^2

85^4 + 85^2 X 204^2 = 18785^2
204^4 + 85^2 X 204^2 = 45084^2
85 + 204 = 17^2

133^4 + 133^2 X 156^2 = 27265^2
156^4 + 133^2 X 156^2 = 31980^2
133 + 156 = 17^2

144^4 + 144^2 X 640^2 = 94464^2
640^4 + 144^2 X 640^2 = 419840^2
144 + 640 = 28^2

184^4 + 184^2 X 345^2 = 71944^2
345^4 + 184^2 X 345^2 = 134895^2
184 + 345 = 23^2

189^4 + 189^2 X 252^2 = 59535^2
252^4 + 189^2 X 252^2 = 79380^2
189 + 252 = 21^2

189^4 + 189^2 X 340^2 = 73521^2
340^4 + 189^2 X 340^2 = 132260^2
189 + 340 = 23^2

217^4 + 217^2 X 744^2 = 168175^2
744^4 + 217^2 X 744^2 = 576600^2
217 + 744 = 31^2

225^4 + 225^2 X 1000^2 = 230625^2
1000^4 + 225^2 X 1000^2 = 1025000^2
225 + 1000 = 35^2

336^4 + 336^2 X 448^2 = 188160^2
448^4 + 336^2 X 448^2 = 250880^2
336 + 448 = 28^2

340^4 + 340^2 X 816^2 = 300560^2
816^4 + 340^2 X 816^2 = 721344^2
340 + 816 = 34^2

400^4 + 400^2 X 561^2 = 275600^2
561^4 + 400^2 X 561^2 = 386529^2
400 + 561 = 31^2

525^4 + 525^2 X 700^2 = 459375^2
700^4 + 525^2 X 700^2 = 612500^2
525 + 700 = 35^2

532^4 + 532^2 X 624^2 = 436240^2
624^4 + 532^2 X 624^2 = 511680^2
532 + 624 = 34^2

820^4 + 820^2 X 861^2 = 974980^2
861^4 + 820^2 X 861^2 = 1023729^2
820 + 861 = 41^2

Paul.