1 Introduction
Among all polynomial sequences, the following two seem to be most distinguished: the sequence of the power maps \(x^{n}\) and the sequence of the Chebyshev polynomials \(T_{n}(x)\), defined either by the property \(T_{n}(\cos (\theta ))=\cos (n\theta )\), or recursively \(T_{0}(x)=1\), \(T_{1}(x)=x\), \(T_{n+2}(x)=2xT_{n+1}(x)-T_{n}(x)\).
Their best known mutual property is probably the composition identity, which we state here for the Chebyshev polynomials:
In this paper we investigate such property – namely, we prove the Chebyshev polynomials analogue of Zsigmondy’s Theorem.
Zsigmondy’s Theorem says for which natural numbers \(a,n{\gt}1\) there is a prime divisor \(p\) of \(a^{n}-1\) that does not divide any of the numbers \(a^{d}-1\), \(d{\lt}n\) (such primes are called primitive prime divisors) or, equivalently, there is a prime number \(p\) such that the multiplicative order \({\mathrm{ord }}_{p}(a)\) equals \(n\).
The above mentioned link between the power maps and the Chebyshev polynomials evoke the question whether we could replace \(a^{n}\) in Zsigmondy’s Theorem by \(T_{n}(a)\). Our answer is Theorem 1; in order to formulate it we have to introduce the following Chebyshevian analogue of the multiplicative order. For an integer \(a\) and a prime number \(p\) denote 1
this quantity always exists by Lemma 3.
Let \(n\) and \(a\) be integers such that \(n{\gt}0\). There exists a prime number \(p\) such that \(n=\text{Che}_{p}(a)\), except in the following cases:
\(a=1\) and \(n{\gt}1\),
\(a=0\) and \(n\notin \left\lbrace 2,4\right\rbrace \),
\(a=-1\) and \(n{\gt}2\),
\(n=1\) and \(a\in \left\lbrace 0,2 \right\rbrace \),
\(n=2\) and \(a=\pm 2^{\alpha -1}-1\),
\(n=3\) and \(a=\tfrac {1}{2}(\pm 3^{\alpha }-1)\),
\(n=4\) and \(a=\pm 2^{\alpha -1}\),
\(n=6\) and \(a=\tfrac {1}{2}(\pm 3^{\alpha }+1)\),
where \(\alpha \) runs through positive integers.