On the class groups of cyclotomic extensions in presence of a solution to Catalan's equation

Abstract

Catalan's conjecture states that the equation x(p)-v(q)=1 has no other integer solutions but 3(2)-2(3)=1. We investigate the consequences of existence of further solutions (with odd prime exponents p, q) upon the relative class group of the pth cyclotomic extension. We thus obtain several new results which merge into the condition q not equivalent to 1 mod p and p not equivalent to 1 mod q. This condition is used in the proof of Catalan's conjecture. (C) 2005 Elsevier Inc. All rights reserved.