Vol. 23, No. 1, 1967

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ISSN: 0030-8730
Addition theorems for sets of integers

Calvin T. Long

Vol. 23 (1967), No. 1, 107–112
Abstract

Let C be a set of integers. Two subsets A and B of C are said to be complementing subsets of C in case every c C is uniquely represented in the sum

C = A + B = {x|x = a + b,a ∈ A,b ∈ B }.

In this paper we characterize all pairs A,B of complementing subsets of

Nn = {0,1,⋅⋅⋅ ,n− 1}

for every positive integer n and show some interesting connections between these pairs and pairs of complementing subsets of the set N of all nonnegative integers and the set I of all integers. We also show that the number C(n) of complementing subsets of Nn is the same as the number of ordered nontrivial factorizations of n and that

        ∑
2C(n) =    C(d).
d|n

Mathematical Subject Classification
Primary: 10.62
Milestones
Received: 15 February 1966
Revised: 22 August 1966
Published: 1 October 1967
Authors
Calvin T. Long