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I have an equation that I think should not have too many solutions, but I don't see a way to argue this.

Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, y \leq N$ can the following equation have.

$$a (x^2 - x) = b(y^2 - y) + c$$

Can we possibly get an asymptotic bound in terms of $N$?

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It may be useful to rewrite your equation as $au^2-bv^2=d$, where $u=2x-1$, $v=2y-1$, and $d=a+c-b$, and then factor the left-hand side. –  Seva May 10 '13 at 18:54
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