Repdigits in Euler functions of Pell and associated pell numbers

TL;DR

This paper proves that the Euler totient function of Pell numbers and certain associated Pell numbers cannot be repdigits with two or more identical digits, highlighting a specific number-theoretic property.

Contribution

It establishes a new result showing the non-existence of multi-digit repdigits in the Euler totient of Pell and related numbers.

Findings

Euler totient of Pell numbers is never a multi-digit repdigit.

The result extends to certain associated Pell numbers.

No Pell number's Euler totient forms a repeated-digit number with at least two digits.

Abstract

A natural number $n$ is called a repdigit if all its digits are same. In this paper, we prove that Euler totient function of no Pell number is a repdigit with at least two digits. This study is also extended to certain subclass of associated Pell numbers.