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ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 91 / Issue 1 / February 2015
- Published online by Cambridge University Press:
- 14 October 2014, pp. 11-18
- Print publication:
- February 2015
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- Article
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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 
$s\geq 1$.
