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

(In reply to re: i cant get it by K Sengupta)

Using:
x^2 + 5y = (x + p)^2, and
y^2 + 5x = (y + q)^2

We see that

5y = 2px + p^2 and
5x = 2qy + q^2

Combining equations gives:

25x = 2q(2px + p^2) + 5q^2

Or:

x = [2qp^2 + 5q^2]/[25 - 4pq]

Since all are positive integers, pq is less than or equal to 6 and so are p and q.

The pairs (p,q) are elements of the set {(1,1),(1,2),(2,2),(1,3),(2,3),(1,4),(1,5),(1,6)} and their mirror images.

Positive integer solutions only exist at (p,q) = (2,2), (2,3), (1,5), and (1,6) and their mirrors.

They are (x,y) = (4,4), (69,56), (11,27), and (192,77) and their mirrors.

Posted by Eric on 2006-06-07 00:10:23