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Bulletin of the London Mathematical Society 2008 40(1):97-107; doi:10.1112/blms/bdm105
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© 2008 London Mathematical Society

Multiplicative decomposability of shifted sets

Christian Elsholtz

Department of Mathematics
Royal Holloway
Egham
Surrey TW20 0EX
United Kingdom

Received 25 May 2007. Revision received 27 July 2007.

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 AB+1subP? 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.

2000 Mathematics Subject Classification 11N25, 11N36 (primary).


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