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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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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.

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