Catalan's equation has no new solution with either exponent less than 10651

Abstract

We consider Catalan's equation $x^p-y^q=1$ (where all variables are integers and $p,q$ are greater than $1$), which has the obvious solution $9-8=1$. Are there others? Applying old and new theoretical results to a systematic computation, we were able to improve on recent work of Mignotte and show that Catalan's equation has only the obvious solutions when $\min\{p,q\}<10651$. Two crucial tools used are a new result of Laurent, Mignotte, and Nesterenko on linear forms of logarithms, and a criterion obtained by W. Schwarz in 1994.