Abstract

A set of Beurling generalized integers consists of the unit n ₀ = 1 plus the set n₁ ≤ n ₂ ≤ … of all power products of a set of generalized primes 1 < g ₁ ≤ g ₂ ≤ g ₃ ≤ … with g_i → ∞, with these power products arranged in increasing order and counted with multiplicity. We say that has the Delone property if there are positive constants r, R such that R ≥ n _{i + 1} – n _i ≥ r for all i ≥ 1. Any set with the Delone property has unique factorization into irreducible elements and is therefore a subsemigroup of ℝ⁺. We classify all such semigroups which are contained in the integers . The set of generalized primes of any such consists of all but finitely many primes, plus finitely many other composites.