July 2016

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

Let n ≥ 1 be an integer. Define Ï•2(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 Ï•2 is a multiplicative function, that is, if gcd(m, n) = 1, then Ï•2(mn) = Ï•2(m)Ï•2(n). Key words: Euler phi-function, multiplicative function.

Author(s): Mbakiso Fix Mothebe and Sebaameng Samuel