Zsigmondy’s Theorem for Chebyshev Polynomials

For an integer \(a\) consider the divisibility sequence \(s_{n}=T_{n}(a)-1\) defined by the Chebyshev polynomials \(T_{n}\). We list all values of \(a\) and \(n\) for which the term \(s_{n}\) has no primitive prime divisor. This result is a pendant to the classic Zsigmondy’s Theorem, an analogous result for power maps.