Multiplicative decomposability of shifted sets

Abstract

The following two problems are open. Do two sets of positive integers 𝒜 and ℬ exist, with at least two elements each, such that 𝒜+ℬ coincides with the set of primes 𝒫 for sufficiently large elements? Let 𝒜={6, 12, 18}. Is there an infinite set ℬ of positive integers such that 𝒜ℬ+1⊂𝒫? A positive answer would imply that there are infinitely many Carmichael numbers with three prime factors. In this paper we prove the multiplicative analogue of the first problem, namely that there are no two sets 𝒜 and ℬ, with at least two elements each, such that the product 𝒜ℬ coincides with any additively shifted copy 𝒫+c of the set of primes for sufficiently large elements. We also prove that shifted copies of sets of integers that are generated by certain subsets of the primes cannot be multiplicatively decomposed.

Locations

• Bulletin of the London Mathematical Society - View
• CiteSeer X (The Pennsylvania State University) - View - PDF