Abstract

In this talk we will study how large sets A and B of integers can be if all elements of their sumset A+B are supposed to have a specified multiplicative form. We also sudy the related problem where shifted elements of product sets have a specified multiplicative form.

Examples: 1) An open problem of Ostmann states that there are no two sets of integers A and B, with at least two elements each such that A+B is (apart from finitely many elements) the set of primes.

This problem is related to the twin prime problem. Let A={0,2}. Is there an infinite set B such that A+B is a subset of the primes?

2) In contrast, the sumset of the set of squares satisfies a multiplicative constraint. 3) We also look at shifted copies of product sets and study for example if the set of shifted primes P-1 can be multiplicatively decomposed.

This is related to another famous problem: Let A={6,12,18}. Is there an infinite set B such that AB+1 is a subset of the primes? This would imply there are infinitely many Carmichael numbers with 3 prime factors.