<s>[INST] Integer solutions to multivariate polynomial I'm looking for a way to show that an equation like  $$15(1+x+x^{2})(1+y+y^{2})=16(xy)^{2}-1$$ has no solutions with x and y odd positive integers. The 16 and 15 can be changed  but I'm trying to generalize it. Any help is appreciated.
 [/INST] Factorization should be useful, but another idea is to note that
if the equation is $a(x^2+x+1)(y^2+y+1)=b(xy)^2-1$ and $a<b$, then $min(x,y)$ is upper-bounded by a function of $a,b$ and it is sufficient to examine those values (and solve for the other).
Specifically, here is one (not necessarily sharp) bound: $min(x,y) \leq \dfrac{\sqrt{b}}{\sqrt{b}-\sqrt{a}}$.
To prove the bound, we write: $a(1+\dfrac{1}{x}+\dfrac{1}{x^2})(1+\dfrac{1}{y}+\dfrac{1}{y^2})=b-\dfrac{1}{x^2y^2}$;
thus $a\dfrac{xy}{(x-1)(y-1)}>b-\dfrac{1}{xy(x-1)(y-1)}$,
and hence $axy>b(x-1)(y-1)-1$, so that $\left(1-\dfrac{1}{x}\right)\left(1-\dfrac{1}{y}\right)\leq\dfrac{a}{b}$.
So if $x,y \geq n$, then $1-1/n \leq \dfrac{\sqrt{a}}{\sqrt{b}}$ and thus $n \leq \dfrac{\sqrt{b}}{\sqrt{b}-\sqrt{a}}$.
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