The Catalan equation

We consider the Catalan equation xp−yq=1 in unknowns x,y,p,q, where x,y are taken from an integral domain A of characteristic 0 that is finitely generated as a Z-algebra and p,q>1 are integers. We give explicit upper bounds for p and q in terms of the defining parameters of A. Our main theorem is a more precise version of a result of Brindza (1993). Brindza (1987) also gave inexplicit bounds for p and q in the special case that A is the ring of S-integers for some number field K. As part of the proof of our main theorem, we will give a less technical proof for this special case with explicit upper bounds for p and q.