On a family of sequences related to Chebyshev polynomials

TL;DR

This paper studies prime occurrences in sequences related to Chebyshev polynomials and proves specific prime distribution properties for certain dilated polynomial values, proposing conjectures for the general case.

Contribution

It establishes prime occurrence patterns in sequences linked to Chebyshev polynomials and introduces conjectures on their infinite prime content for most parameter values.

Findings

Prime occurrence when n is a dilated Chebyshev polynomial of the first kind

Sequences contain at most one prime for specific polynomial values of n

Conjecture: sequences have infinitely many primes for other values of n

Abstract

The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$) dilated versions of the so-called Chebyshev polynomials of the fourth kind, also known as airfoil polynomials. It is proved that when the value of $n$ is given by a dilated Chebyshev polynomial of the first kind evaluated at a suitable integer, either the sequence contains a single prime, or no term is prime. For all other values of $n$, it is conjectured that the sequence contains infinitely many primes, whose distribution has analogous properties to the distribution of Mersenne primes among the Mersenne numbers. Similar results are obtained for the sequences associated with negative integers $n$, which correspond to Chebyshev polynomials of the third kind, and to another family of Lehmer numbers.