If (x^2+y^2+x)=2kxy, k>0, then by adding 2xy-x to both sides and factoring, one gets (x+y)^2=((2k+2)y+1)x
If a prime p divides x, then it must divide the left side x^2+2xy+y^2, and so p divides y^2 and thus y.
But then this implies ((2k+2)y+1) is not divisible by p, and thus since (x+y)^2 must have an even power of p in its prime factorization, so must x.
Since this applies for every prime p dividing x, this implies x must be the square of some integer.
Posted by Gamer
on 2011-03-08 03:53:11