The index of small length sequences

David J. Grynkiewicz, Uzi Vishne

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let n ≥ 2 be a fixed integer. Define (x)n to be the unique integer in the range 0 ≤ (x)n < n which is congruent to x modulo n. Given x1,⋯,x, let (x1,⋯,x)Z1 = min{(ux1)n + ⋯ + (uxR)n:u Z;, gcd(u,n) = 1} and define Ind(x1,⋯,xR) = 1 nZ(x1,⋯,xR)Z1 to be the index of the sequence (x1,⋯,xR). If x1,⋯,x4 have α [1,4]xα 0 (modn) but α Ixα 0 for all proper, non-empty subsets I [1, 4], then a still open conjecture asserts that Ind(S) = 1 provided that gcd(n, 6) = 1. We give an alternative proof, that does not rely on computer calculations, verifying this conjecture when n is a product of two prime powers.

Original languageEnglish
Pages (from-to)977-1014
Number of pages38
JournalInternational Journal of Algebra and Computation
Volume30
Issue number5
DOIs
StatePublished - 1 Aug 2020

Bibliographical note

Publisher Copyright:
© 2020 World Scientific Publishing Company.

Keywords

  • Zero-sum
  • block monoid
  • index
  • subsequence sum
  • subsum
  • zero-sum free

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