Abstract

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.

Highlights

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

  • Let n ≥ 1 be an integer and let S = {1, 7, 11, 13, 17, 19, 23, 29}, the set of integers which are both less than and relatively prime to 30

  • The primary goal of this study is to show that φ2 is a multiplicative function, that is, if gcd(m, n) = 1, φ2(mn) = φ2(m)φ2(n)

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Summary

Introduction

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

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