Pell Numbers whose Euler Function is a Pell Number

TL;DR

This paper proves that among Pell numbers, only 1 and 2 have the property that their Euler function results in another Pell number, highlighting a unique numerical relationship.

Contribution

The paper establishes a complete characterization of Pell numbers with Pell Euler functions, identifying only two such numbers.

Findings

Only Pell numbers 1 and 2 have Pell Euler functions.

The Euler function of other Pell numbers is not a Pell number.

The result narrows the understanding of Pell numbers and their Euler functions.

Abstract

In this paper, we show that the only Pell numbers whose Euler function is also a Pell number are $1$ and $2$.