Diophantine equations involving Euler’s totient function
Volume 191 / 2019
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 $a \gt b\ge 1$.
