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ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

Part of: Diophantine equations

Published online by Cambridge University Press:  14 October 2014

NOBUHIRO TERAI
NOBUHIRO TERAI*
Affiliation:
Department of Computer Science and Intelligent Systems, Faculty of Engineering, Oita University, 700 Dannoharu, Oita 870-1192, Japan email [email protected]
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Abstract

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Let $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ 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 the equation has solutions, then $m=p^{s}$ or $m=2p^{s}$ with $p$ an odd prime and $s\geq 1$.

Keywords

exponential Diophantine equationsFermat quotientEuler quotientinteger solutions

MSC classification

Primary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 11 - 18
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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