2 Preliminary results

Proposition 2

✓

If \(a,b\) are nonnegative integers, then

\[ (T_{a+b}(x)-1)(T_{\left|a-b \right| }(x)-1)=(T_{a}(x)-T_{b}(x))^2. \]

The following lemma is the analogue of Fermat’s little theorem for Chebyshev polynomials.

Lemma 3

✓

Let \(p\) be an odd prime number. For every integer \(x\)

\[ T_{p-1}(x)\equiv 1 \mod p \, \, \, \text{or} \, \, \, T_{p+1}(x)\equiv 1 \mod p. \]

For every integer \(x\)

\[ T_{2}(x)\equiv 1 \mod 2. \]

Lemma 4

✓

Let \(p\) be a prime number and \(x\) be an integer. Let \(m\) be the smallest positive integer such that \(T_{m}(x) \equiv 1 \mod p\). Then \(T_{n}(x) \equiv 1 \mod p\) for a positive integer \(n\) if and only if \(m \mid n\).

Lemmas 3 and 4 together with \(T_{1}(x)=x\) give the following.

Lemma 5

✓

If \(x\) is an integer and \(p\) is an odd prime number then \(\text{Che}_{p}(x)\) divides \(p-1\) or \(p+1\). In particular, \(\text{Che}_{p}(x)\) and \(p\) are coprime. If \(x\) is odd then \(\text{Che}_{2}(x)=1\). If \(x\) is even then \(\text{Che}_{2}(x)=2\).

Lemma 6

✓

For every \(n\ge 1\)

\[ T_{n}(x)-1=\prod _{d \mid n}\Omega _{d}^{\sigma _{d}}(x) \]

where \(\Omega _{1}(x)=x-1\) and for \(d\ge 2\)

\[ \Omega _{d}(x)=\prod _{ \substack { 1\le k \le \frac{d}{2}\\ \gcd (k,d)=1}}2(x-\cos \tfrac {2 k \pi }{d}) \]

and

\[ \sigma _{d}= \left\{ \begin{array}{rl} 1 & \text{if } d=1,2,\\ 2 & \text{if } d {\gt}2. \end{array} \right. \]

Proposition 7

✓

Let \(m, n\) be positive integers. Then the polynomial \(\Omega _{mn}\) is a divisor of \(\Omega _{n}(T_{m}(x))\). If moreover \(n\ge 3\) and every prime divisor of \(m\) divides \(n\), then \(\Omega _{mn}=\Omega _{n}(T_{m}(x))\).

Proposition 8

✓

Let \(n\) be an odd natural number. Then \(\Omega _{n}(0)=\pm 1\). If moreover \(n\ge 3\) then \(\Omega _{2n}(x)=\pm \Omega _{n}(-x)\).

Proposition 9

✓

Let \(\mathbb {K}\) be a field of characteristic \(0\). Suppose that \(P\in \mathbb {K}[x]\), \(P(0)=\pm 1\), and \(P^{2}\in {\mathbb {Z}}[x]\). Then \(P\in {\mathbb {Z}}[x]\).

Lemma 10

✓

\(\Omega _{n}\in {\mathbb {Z}}[x]\).

Corollary 11

✓

Let \(x\) be an integer. Every primitive prime divisor of \(T_{n}(x)-1\) divides \(\Omega _{n}(x)\).

Proposition 12

✓

For every natural number \(n\) and every nonzero real number \(x\)

\[ \frac{T_{n}(x+1)-1}{x}=n^2+\frac{n^2(n^2-1)}{6}x+\frac{n^2(n^2-1)(n^2-4)}{90}x^{2}+\ldots \in {\mathbb {Z}}[x], \]

where the dots denote irrelevant terms.