Lower bounds for Catalan's equation (Q1127597)

It is conjectured that Catalan's equation \[ x^p - y^q = 1, \quad p \text{ and } q \text{ prime, } \quad x \cdot y \not= 0 \] has only the solution \( (x, y, p, q) = (3, 2, 2, 3). \) Using Baker's results on linear forms in logarithms, \textit{R. Tijdeman} [Acta Arith. 29, 197-209 (1976; Zbl 0286.10013)] showed that there are no solutions if \( \max (p, q) \) is large. By extensive calculations, based on a criterion of Inkeri (in an improved version by Schwarz) on divisibility properties of the relative class number \( h^{-}(K_p) \), where \( K_p \) is a certain subfield of the cyclotomic field \( {\mathbb Q}(e^{2\pi i/p})\), and on an elementary criterion of Mignotte, the authors show: For \(\min(p, q)< 30000\), Catalan's equation has no solution \(\not= (3,2,2,3)\).