Sequences of integers generated by two fixed primes

Abstract Let p and q be two distinct fixed prime numbers and (nᵢ) ₈ ₀ the sequence of consecutive integers of the form pᵃ qᵇ with a, b 0. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size n₈+₁-nᵢ, with each bound containing an unspecified exponent and implicit constant. We will explicitly bound these four quantities. Earlier Langevin (1976) gave weaker estimates for (only) the exponents. Given a real number >1, there exists a smallest number m such that for every n m, there exists an integer nᵢ in [n, n). Our effective version of Tijdeman’s result immediately implies an upper bound for m, which using the Koksma–Erdős–Turan inequality we will improve on. We present a fast algorithm to determine m when \p, q\ is not too large and demonstrate it with numerical material. In an appendix we explain, given nᵢ, how to efficiently determine both n₈-₁ and n₈+₁, something closely related to work of Bérczes, Dujella and Hajdu.