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.