= Exponent bound for a binomial power equation
{title2=$ax^n-by^n=c\ne0,\qquad\max(x,y)>1$}
For fixed positive $a,b$ and fixed nonzero $c$, with $x\ne y$, put $Y=\max(x,y)$ and take $\Lambda=\log(a/b)+n\log(x/y)$, swapping the variables and coefficients if necessary. The equation gives $\log|\Lambda|\leq-n\log Y+O(1)$ when the larger-variable power is sufficiently large. A two-logarithm lower bound gives $\log|\Lambda|\geq-C(a,b)\log Y\log(2n)$. Hence $n\leq C\log(2n)+O(1)$, so $n$ is bounded. The cases $x=y$ and a bounded larger-variable power are elementary. For each remaining exponent, the <Thue theorem> for a binomial form supplies finitely many pairs. The constant-power exception $x=y=1$, $a-b=c$, must be removed.
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