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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