Let \((g_i)_{i \geq 1}\) be a sequence of Chebyshev polynomials, each with degree at least two, and define \((f_i)_{i \geq 1}\) by the following recursion: \(f_1=g_1\), \(f_n=g_n\circ f_{n-1}\) for \(n \geq2\). Choose \(\alpha \in \mathbb{Q}\) such that \(\{ g_1^n (\alpha) : n \geq 1 \}\) is an infinite set. The main result of this talk is as follows: If \(f_n(\alpha)=\frac{A_n}{B_n}\) is written in lowest terms, then for all but finitely many \(n>0\) the numerator \(A_n\) has a primitive divisor; that is, there is a prime \(p\) which divides \(A_n\) but does not divide \(A_i\) for any \(i \lt n\). I will also talk about some further directions of my future work at the end.
Primitive Divisors in Generalized Iterations of Chebyshev Polynomials
Primitive Divisors in Generalized Iterations of Chebyshev Polynomials