Diophantine equations involving Euler's totient function

TL;DR

This paper investigates Diophantine equations involving Euler's totient function and Lucas sequences, proving the absence of solutions in positive integers except for specific trivial cases.

Contribution

It establishes new non-existence results for certain equations involving Euler's totient function and Lucas sequences, identifying all trivial solutions.

Findings

No solutions for $\phi (x^m-y^m)=x^n-y^n$ except trivial cases.

No solutions for $\phi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ except trivial cases.

Characterization of trivial solutions in these equations.

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 $a>b\ge 1$.