On relatively prime subsets, combinatorial identities, and diophantine equations

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5 Citations (Scopus)

Abstract

Let n be a positive integer and let A be a nonempty finite set of positive integers. We say that A is relatively prime if gcd(A) = 1, and that A is relatively prime to n if gcd(A, n) = 1. In this work we count the number of nonempty subsets of A that are relatively prime and the number of nonempty subsets of A that are relatively prime to n. Related formulas are also obtained for the number of such subsets having some fixed cardinality. This extends previous work for the case where A is an interval of successive integers. As an application we give some identities involving Möbius and Mertens functions, which provide solutions to certain Diophantine equations.

Original languageEnglish
JournalJournal of Integer Sequences
Volume15
Issue number3
Publication statusPublished - Mar 25 2012

Keywords

  • Combinatorial identities
  • Diophantine equation
  • Mertens function
  • Möbius function
  • Phi function
  • Relatively prime set

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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