On the Euler function of linearly recurrence sequences

TL;DR

This paper proves that for most integers n, the Euler function of the absolute value of a nondegenerate linearly recurrent sequence exceeds the sequence value at the Euler function of n, with the failure set being very sparse.

Contribution

It establishes a new inequality relating Euler functions and linearly recurrent sequences, showing it holds for almost all n with a very small exceptional set.

Findings

The inequality holds for a set of positive integers with density 1.

The set where the inequality fails has size O(x / log x).

The result applies to nondegenerate sequences not polynomial in n.

Abstract

In this paper, we show that if $(U_n)_{n\ge 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$, then the inequality $\phi(|U_n|)\ge |U_{\phi(n)}|$ holds on a set of positive integers $n$ of density $1$, where $\phi$ is the Euler function. In fact, we show that the set of $n\le x$ for which the above inequality fails has counting function $O_U(x/\log x)$.