Sequences of integers generated by two fixed primes

Abstract

Let p and q be two distinct fixed prime numbers and \((n_i)_{i\ge 0}\) the sequence of consecutive integers of the form \(p^a\cdot q^b\) with \(a,b\ge 0\). Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size \(n_{i+1}-n_i\), 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 \(\alpha >1\), there exists a smallest number m such that for every \(n\ge m\), there exists an integer \(n_i\) in \([n,n\alpha )\). 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 \(\max \{p,q\}\) is not too large and demonstrate it with numerical material. In an appendix we explain, given \(n_i\), how to efficiently determine both \(n_{i-1}\) and \(n_{i+1}\), something closely related to work of Bérczes, Dujella and Hajdu.