Catalan’s equation 𝑥^{𝑝}-𝑦^{𝑞}=1 and related congruences

Abstract

Abstract.We investigate solutions of Catalan's equation x -yq = 1 in nonzero integers x, y, p, q . By use of class numbers and congruences pq = p (mod g ) we show the impossibility of the equation for a large number of prime exponents p , q . Both theoretical and computer results are included. We also study lower bounds on possible, hitherto unknown, solutions x, y > 2; we especially wish to communicate the bound x , y > 10