Partitioning integers using a finitely generated semigroup

Daniel Reich

Vol. 78 (1978), No. 1, 233–240

DOI: 10.2140/pjm.1978.78.233

Abstract

Denoting by Γ the semigroup of positive integers generated by two fixed primes, let r_k(N) be the number of partitions of N as a sum of k elements of Γ. Our main result is that r₂(N) is a bounded function of N. Incidentally, we obtain an estimate of the number of distinct prime divisors of numbers of the form 1 + qⁿ. Boundedness of r_k(N) would resolve an approximation theoretic conjecture of D. J. Newman.

Mathematical Subject Classification 2000

Primary: 10A05, 10A05

Secondary: 30E10, 10A45

Milestones

Received: 4 November 1977

Published: 1 September 1978