Sequences of integers generated by two fixed primes

Abstract

Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ 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-Erd\H{o}s-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 B\'erczes, Dujella and Hajdu.