Sequences of integers generated by two fixed primes

Let p and q be two distinct fixed prime numbers and (nᵢ)i≥ 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 nᵢ₊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 α >1 , there exists a smallest number m such that for every n≥ m , there exists an integer nᵢ in [n,nα ) . 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 ,q\ is not too large and demonstrate it with numerical material. In an appendix we explain, given nᵢ , how to efficiently determine both nᵢ₋₁ and nᵢ₊₁ , something closely related to work of Bérczes, Dujella and Hajdu.