Czechoslovak Mathematical Journal, Vol. 58, No. 1, pp. 51-59, 2008

On the Euler function of repdigits

Florian Luca

Florian Luca, Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, Mexico, e-mail: fluca@matmor.
unam.mx

Abstract: For a positive integer $n$ we write $\phi(n)$ for the Euler function of $n$. In this note, we show that if $b>1$ is a fixed positive integer, then the equation
\phi\Big(x\frac{b^n-1}{b-1}\Big)=y\frac{b^m-1}{b-1},\qquad{\text where} \^^Mx, y\in\{1,\ldots,b-1\},
has only finitely many positive integer solutions $(x,y,m,n)$.

Keywords: Euler function, prime, divisor

Classification (MSC 2000): 11A25


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