On the shortest weakly prime-additive numbers

Abstract

TextA positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p1,…,pt of n and positive integers α1,…,αt such that n=p1α1+⋯+ptαt. It is clear that t≥3. In 1992, Erdős and Hegyvári proved that, for any prime p, there exist infinitely many weakly prime-additive numbers with t=3 which are divisible by p. In this paper, we prove that, for any positive integer m, there exist infinitely many weakly prime-additive numbers with t=3 which are divisible by m if and only if 8∤m. We also present some related results and pose several problems for further research. VideoFor a video summary of this paper, please visit https://youtu.be/WC_VRFtY07c.