Diophantine equations involving Euler function

TL;DR

This paper investigates specific Diophantine equations involving Euler's totient function, proving the absence of nontrivial solutions except for certain explicit cases, thereby advancing understanding of the solutions' structure.

Contribution

The paper establishes the nonexistence of nontrivial solutions for two classes of Diophantine equations involving Euler's function, identifying all solutions explicitly.

Findings

No nontrivial solutions for the first equation except specified solutions.

No nontrivial solutions for the second equation except specified solutions.

Explicit characterization of solutions involving powers and Euler's function.

Abstract

In this paper, we show that the equation $\varphi(|x^{m}-y^{m}|)=|x^{n}-y^{n}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\neq0, m>0, n>0$ except for the solutions $(x,y,m,n)=((2^{t-1}\pm1),-(2^{t-1}\mp1),2,1), (-(2^{t-1}\pm1),(2^{t-1}\mp1),2,1),$ where $t$ is a integer with $t\geq 2.$ The equation $\varphi(|\frac{x^{m}-y^{m}}{x-y}|)=|\frac{x^{n}-y^{n}}{x-y}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\neq0, m>0, n>0$ except for the solutions $(x,y,m,n)=(a\pm1, -a, 1, 2), (a\pm i, -a, 2, 1),$ where $a$ is a integer with $i=1,2.$