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arXiv · 1711.00180

Diophantine equations involving Euler's totient function

Yong-Gao Chen·Hao Tian
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Abstract

In this paper, we consider the 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 solutions $(x, y, m , n)=(a+1, a, 1, 1)$, 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 solutions $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a>b\ge 1$.

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Yong-Gao Chen, Hao Tian. 2017-11-01. Diophantine equations involving Euler's totient function. https://doi.org/10.4064/aa180402-12-12

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