We consider Catalan's equation xp − yq = 1, where p and q are odd primes, p < q, and x, y integers with |x|, |y| > 1. We introduce a new linear form of logarithms which leads to some improvement of the general upper bounds on p and q for p ??? 3 (mod 4). We also prove that Catalan's equation has no solution when p = 19 and that, when p = 53, the only possible value for q is q = 4889.
