Vol. 53, No. 2, 1974

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Arithmetic properties of certain recursively defined sets

David Anthony Klarner and R. Rado

Vol. 53 (1974), No. 2, 445–463
Abstract

Let R denote a set of linear operations defined on the set N of nonnegative integers; for example, a typical element of R has t he form ρ(x1,,xr) = m0 + m1x1 + + mrxr where m0,,mr denote certain integers. Given a set A of positive integers, there is a smallest set of positive integers denoted R : Awhich contains A as a subset and is closed under every operation in R. The set R : Acan be constructed recursively as follows: Let A0 = A, and define

Ak+1 = Ak ∪ {ρ(a) : ρ ∈ R,a-∈ Ak × ⋅⋅⋅× Ak}(k = 0,1,⋅⋅⋅ ),

then it can be shown that R : A= A0 A1 . The sets R : Asometimes have an elegant form, for example, the set 2x + 3y : 1consists of all positive numbers congruent to 1 or 5 modulo 12. The objective is to give an arithmetic characterization of elements of a set R : A. This paper is a report on progress made on this problem when the authors collaborated at Reading University in the academic year 1970–71.

Mathematical Subject Classification 2000
Primary: 10A99
Secondary: 05B10, 04A05
Milestones
Received: 29 November 1972
Revised: 21 January 1974
Published: 1 August 1974
Authors
David Anthony Klarner
R. Rado