African Journal of
Mathematics and Computer Science Research

  • Abbreviation: Afr. J. Math. Comput. Sci. Res.
  • Language: English
  • ISSN: 2006-9731
  • DOI: 10.5897/AJMCSR
  • Start Year: 2008
  • Published Articles: 225

Short Communication

Properties of the Euler phi-function on pairs of positive integers (6x - 1, 6x + 1)

Mbakiso Fix Mothebe*
  • Mbakiso Fix Mothebe*
  • Department of Mathematics, University of Botswana, Pvt Bag 00704, Gaborone, Botswana.
  • Google Scholar
Sebaameng Samuel
  • Sebaameng Samuel
  • Department of Mathematics, University of Botswana, Pvt Bag 00704, Gaborone, Botswana.
  • Google Scholar


  •  Received: 25 April 2016
  •  Accepted: 20 July 2016
  •  Published: 31 July 2016

Abstract

Let n ≥ 1 be an integer. Define ϕ2(n) to be the number of positive integers x, 1 ≤ x ≤ n, for which both 6x−1 and 6x+1 are relatively prime to 6n. The primary goal of this study is to show that ϕ2 is a multiplicative function, that is, if gcd(m, n) = 1, then ϕ2(mn) = ϕ2(m)ϕ2(n).

Key words: Euler phi-function, multiplicative function.