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.