On a Family of Sequences Related to Chebyshev Polynomials

We consider the appearance of primes in a family of linear recurrence sequences

labelled by a positive integer n. 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.

We prove 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, we conjecture 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.