Number of solution of a diophantine

Question

The size of the set ${ (x, y) ∈ Z × Z : 1/x + x / y + 253 / xy = 1}$ is

Evaluating, we have :$x^2-xy+y=253$ , I am now currently stuck as i cant factor the left hand side.

Answer 1

From $x^2 - xy + y = -253$, $x^2 - xy + \frac 14 y^2 - \frac14 y^2 + y = -253$, $(x-\frac 12 y)^2 - (\frac12 y - 2)^2 = -254$. Let $x' = 2x - y$ and $y' = y-4$, then $x'^2 - y'^2 = -8\times 127$, so $(x' + y')(x' - y') = -8\times 127$. By suitable factoring of $8\times 127$ you should get 4 systems of equations and hence 4 solutions.

Answer 2

Hint: $$y=\dfrac{x^2+253}{x-1}=x+1+\dfrac{254}{x-1}$$