On the exponential Diophantine equation containing the Euler quotients

N. Terai

On the exponential Diophantine equation containing the Euler quotients

Authors

N. Terai Ashikaga Institute of Technology

Keywords:

exponential Diophantine equation, Fermat quotients, Euler quotients, integer solutions

Abstract

Let

$a$ and

$m$ be relatively prime positive integers with

$a>1$ and

$m>2$ . Let

$\phi(m)$ be Euler's totient function. The quotient

$E_m(a)= \frac{~a^{\phi(m)}-1~}{m}$ is called the

$Euler$

$quotient$ of

$m$ with base

$a$ . By Euler's theorem,

$E_{m}(a)$ is an integer. In this paper, we consider the Diophantine equation

$E_{m}(a)=x^l ~(*)$ in integers

$x>1,l>1$ . We conjecture that this equation has exactly five solutions

$(a,m,x,l)$ except for

$(l,m)=(2,3),(2,6)$ , and show that if equation

$(*)$ has solutions, then

$m=p^s$ or

$m=2p^s$ with

$p$ odd prime and

$s \geq 1$ . DOI: 10.1017/S0004972714000719

Published

2014-11-13

Issue

Vol. 91 No. 1 (2015)

Section

Articles

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