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Conditions for a Zero Sum Modulo n

Published online by Cambridge University Press:  20 November 2018

J. D. Bovey
Affiliation:
University of York
Paul Erdös
Affiliation:
University of Colorado
Ivan Niven
Affiliation:
University of Oregon
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In this paper the following result is proved.

Let n > 0 and k ≥ 0 be integers with n — 2k ≥ 1. Given any n — k integers

there is a non-empty subset of indices I ⊂ {1, 2,…, n — k} such that the sum Σi∊I ≡ 0(mod n) if at most n — 2k of the integers (1) lie in the same residue class modulo n.

The result is best possible if n ≥ 3k — 2 in the sense that if "at most n — 2k" is replaced by "at most n — 2k + 1" the result becomes false. This can be seen by taking aj = 1 for 1 ≤ j ≤ n — 2k + 1 and aj = 2 for n —2k + 2 ≤ j ≤ n — k, noting that the number of 2's here is n — k — (n — 2k + 1)= k — 1 ≤ n — 2k + 1.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Halberstarn, H. and Roth, K. F., Sequences I, Oxford, 1966, p. 50, Theorem 15'.Google Scholar
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