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.