Zsigmondy's Theorem for Chebyshev polynomials

TL;DR

This paper investigates the properties of Chebyshev polynomial sequences, specifically identifying all pairs where the sequence terms lack primitive prime divisors, extending classical number theory results.

Contribution

It provides a complete classification of pairs (n, a) where T_n(a) - 1 has no primitive prime divisor, applying Zsigmondy's theorem to Chebyshev polynomials.

Findings

Identifies all pairs (n, a) with no primitive prime divisor in T_n(a) - 1

Extends Zsigmondy's theorem to Chebyshev polynomial sequences

Provides a comprehensive list of exceptional cases

Abstract

For an integer $a$ consider the sequence $(T_{n}(a)-1)_{n=1}^{\infty}$ defined by the Chebyshev polynomials $T_{n}$. We list all pairs $(n,a)$ for which the term $T_{n}(a)-1$ has no primitive prime divisor.