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Diophantine equations involving Euler’s totient function

Volume 191 / 2019

Yong-Gao Chen, Hao Tian Acta Arithmetica 191 (2019), 33-65 MSC: Primary 11A25; Secondary 11D61, 11D72. DOI: 10.4064/aa180402-12-12 Published online: 6 August 2019

Abstract

We consider equations involving Euler’s totient function $\phi $ and Lucas type sequences. In particular, we prove that the equation $\phi (x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the trivial $(x, y, m , n)=(a+1, a, 1, 1)$, where $a$ is a positive integer, and the equation $\phi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ has no solutions in positive integers $x, y, m, n$ except for the trivial $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a \gt b\ge 1$.

Authors

  • Yong-Gao ChenSchool of Mathematical Sciences and Institute of Mathematics
    Nanjing Normal University
    Nanjing 210023, P.R. China
    e-mail
  • Hao TianSchool of Mathematical Sciences and Institute of Mathematics
    Nanjing Normal University
    Nanjing 210023, P.R. China
    e-mail

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