Let $\ell$ be a prime number, $F$ be a global function field of characteristic $\ell$. Assume that there is a prime $P_\infty$ of degree $1$. Let $\mathcal{ O}_F$ be the ring of functions in $F$ with no poles outside of $\{ P_\infty\}$. We study solutions to Catalan's equation $X^m-Y^n=1$ over $\mathcal{ O}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in $\mathcal{ O}_F$, when $m,n>1$.
Categories: math.NT