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On the Euler function of repdigits

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Abstract

For a positive integer n we write φ(n) for the Euler function of n. In this note, we show that if b > 1 is a fixed positive integer, then the equation

$$ \varphi \left( {x\frac{{b^n - 1}} {{b - 1}}} \right) = y\frac{{b^m - 1}} {{b - 1}}, where x,y \in \{ 1, \ldots ,b - 1\} , $$

has only finitely many positive integer solutions (x, y, m, n).

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, C. P. 58089, Morelia, Michoacán, México

    Florian Luca

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  1. Florian Luca
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Correspondence to Florian Luca.

Additional information

Dedicated to William D. Banks on his $$ \sqrt {\varphi (2005)} ^{th} $$ birthday.

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Luca, F. On the Euler function of repdigits. Czech Math J 58, 51–59 (2008). https://doi.org/10.1007/s10587-008-0004-0

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  • DOI: https://doi.org/10.1007/s10587-008-0004-0

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