A Small Contribution to Catalan′s Equation Original Research Article

Using recent results on linear forms in logarithms of algebraic numbers, we prove that any solution of the equation xp − yq = ε, where ε = ± 1, p and q are odd primes, and p > q satisfies p < 3.42 · 1028 and q < 5.6 · 1019. We also combine our work with some results of Altonen and Inkeri to determine the six cases with q ≤ 37 for which this equation may have solutions.