Abstract

Using recent results on linear forms in logarithms of algebraic numbers, we prove that any solution of the equation x p − y q = ϵ, where ϵ = ± 1, p and q are odd primes, and p > q satisfies p < 3.42 · 10 28 and q < 5.6 · 10 19. 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.

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