Find all possible pairs of positive integers (x,y) such that both x²+5y and y²+5x are perfect squares.

Let's suppose x≠y. If x²+5y=p² and y²+5x=q², then p²-q²= (x²-y²) -5(x-y)= (x-y)(x+y-5); note that for all integer x and y, one of this factors is odd, and the other is even, so the product is a multiple of 4, plus 2.

However, p²-q²= (p+q)(p-q), and for all integer p and q, those two factors are both odd or both even, so the product is either a multiple of 4, or a multiple of 4, plus 1.

In conclusion, there can be no answer for x≠y.

Edited on June 3, 2006, 10:24 am

Posted by e.g. on 2006-06-03 10:22:46