On a family of sequences related to Chebyshev polynomials

Abstract:

The appearance of primes in a family of linear recurrence sequences labelled by a positive integer nn is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in nn) 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 nn 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 nn, 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 nn, which correspond to Chebyshev polynomials of the third kind, and to another family of Lehmer numbers.