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I'm trying to prove that there exists two distinct positive integers whose sum and difference are both perfect squares. I cannot find any pattern or characteristic between the pairs of numbers that work i.e.

  • 4, 5
  • 6, 10
  • 8, 17
  • 10, 6
  • 10, 26

Any help will be greatly appreciated! Thanks!

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Let those positive integers be $a$ and $b$.Then,

$a+b=x^2$

$a-b=y^2$

Adding the two we get $a=\frac{x^2+y^2}{2}$ and $b=\frac{x^2-y^2}{2}$

Since, a and b are integers we must have $x^2+y^2$ and $x^2-y^2$ must be even, for that we must have x and y both even or both odd.

Now for finding such pairs take any even $x,y$ for example let x=8 and y=4

which gives $a=40$ and $b=24$, we have $a+b=64=8^2$ and $a-b=16=4^2$

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  • $\begingroup$ Ramanujan would be proud (+1) $\endgroup$ – J. W. Perry Oct 30 '13 at 4:37
  • $\begingroup$ @J.W.Perry Thank you,appreciated. $\endgroup$ – Shobhit Oct 30 '13 at 4:39
  • $\begingroup$ This proof is great! Well done! :) $\endgroup$ – 1233dfv Oct 30 '13 at 4:53
  • $\begingroup$ @RossBelgram Thank you, appreciated. $\endgroup$ – Shobhit Oct 30 '13 at 4:53

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