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

Mbakiso Fix Mothebe*, Sebaameng Samuel

Research output: Contribution to journalArticle

### 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.
Original language English 12-14 3 African Journal of Mathematics and Computer Science Research 9 2 https://doi.org/10.5897/AJMCSR2016.0648 Published - 2016

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Euler's phi function
Multiplicative Functions
Relatively prime
Integer

### Cite this

@article{5dc2c63098e04f76aea8648f27ec0222,
title = "Properties of the Euler phi-function on pairs of positive integers (6x - 1, 6x + 1)",
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.",
author = "Mothebe*, {Mbakiso Fix} and Sebaameng Samuel",
year = "2016",
doi = "10.5897/AJMCSR2016.0648",
language = "English",
volume = "9",
pages = "12--14",
journal = "African Journal of Mathematics and Computer Science Research",
issn = "2006-9731",
number = "2",

}

In: African Journal of Mathematics and Computer Science Research, Vol. 9, No. 2, 2016, p. 12-14.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Properties of the Euler phi-function on pairs of positive integers (6x - 1, 6x + 1)

AU - Mothebe, Mbakiso Fix

AU - Samuel, Sebaameng

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

U2 - 10.5897/AJMCSR2016.0648

DO - 10.5897/AJMCSR2016.0648

M3 - Article

VL - 9

SP - 12

EP - 14

JO - African Journal of Mathematics and Computer Science Research

JF - African Journal of Mathematics and Computer Science Research

SN - 2006-9731

IS - 2

ER -