Generalized iterations and primitive divisors

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• 作者：Nathan Wakefield ^{nathan.wakefield@unl.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primitive divisors ; Arithmetic dynamics ; Chebyshev polynomials
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：159
• 期：Complete
• 页码：273-294
• 全文大小：389 K

文摘

Let (g_i)_i≥1${(g_i)}_{i\ge 1}$ be a sequence of Chebyshev polynomials, each with degree at least two, and define (f_i)_i≥1${(f_i)}_{i\ge 1}$ by the following recursion: f₁=g₁$f_1=g_1$, f_n=g_n鈭榝_n−1, for n≥2$n\ge 2$. Choose 伪∈Q$伪\in \mathbb{Q}$ such that $\{g_1^n(伪):n\ge 1\}$ is an infinite set. The main result is as follows: If $f_n(伪)=\frac{A_n}{B_n}$ is written in lowest terms, then for all but finitely many n>0$n>0$, the numerator, A_n$A_n$, has a primitive divisor; that is, there is a prime p   which divides A_n$A_n$ but does not divide A_i$A_i$ for any i<n$i<n$.