Multiplicative decomposability of shifted multiplicatively defined sets
by
Christian Elsholtz
Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, UK

The following two problems are open:

1) 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?

2) Let A={6,12,18}. Is there an infinite set B of positive

integers such that all elements of the shifted product set

AB+1 are prime? A positive answer would imply that there are infinitely many Carmichael

numbers with 3 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 which are

generated by certain subsets of the primes cannot be multiplicatively

decomposed.

Date received: May 26, 2008