Geodesic
On the Euler function of repdigits
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 51-59 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 \[ \phi \Big (x\frac{b^n-1}{b-1}\Big )=y\frac{b^m-1}{b-1},\qquad {\text{where}} \ x,~y\in \lbrace 1,\ldots ,b-1\rbrace , \] has only finitely many positive integer solutions $(x,y,m,n)$.
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 \[ \phi \Big (x\frac{b^n-1}{b-1}\Big )=y\frac{b^m-1}{b-1},\qquad {\text{where}} \ x,~y\in \lbrace 1,\ldots ,b-1\rbrace , \] has only finitely many positive integer solutions $(x,y,m,n)$.
Classification : 11A25
Keywords: Euler function; prime; divisor 
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Luca, Florian. On the Euler function of repdigits. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 51-59. http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a3/

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