Vol. 53, No. 2, 1974

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 331: 1
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
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