On a Family of Sequences Related to Chebyshev Polynomials

Hone, Andrew N.W., Jeffery, L. Edson, Selcoe, Robert G. (2018) On a Family of Sequences Related to Chebyshev Polynomials. Journal of Integer Sequences, 21 . 18.7.2. ISSN 1530-7638.

Abstract

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.

Item Type:	Article
Subjects:	Q Science > QA Mathematics (inc Computing science)
Divisions:	Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User:	Andrew N W Hone
Date Deposited:	10 Oct 2018 14:15 UTC
Last Modified:	29 May 2019 21:16 UTC
Resource URI:	https://kar.kent.ac.uk/id/eprint/69496 (The current URI for this page, for reference purposes)
Hone, Andrew N.W.:	https://orcid.org/0000-0001-9780-7369