A class number free criterion for catalanʹs conjecture Original Research Article

The Catalan conjecture asserts that the equation XU−YV=1 with U,V>1 has no other solution in integers but 32−23=1 (Catalanʹs Conjecture, Academic Press, New York, 1994). We prove that, for primes U=p and V=q yielding a solution to the Catalan equation, the simultaneous conditionsimagepq−1≡1mod q2,imageqp−1≡1mod p2must necessarily hold. The proof is a variation, using Stickelberger elements, of a technical theme originated by Inkeri, Mignotte and continued by Schwarz, each with different points of focus.