Teng Cheng: Primitive prime divisors for weighted homogeneous polynomial, 173-182

Abstract:

From Mersenne sequence to elliptic divisibility sequence, it is a important problem in number theory to prove the existence of primitive prime divisors of an arithmetically defined sequence, i.e., the finiteness of the relevant Zsigmondy set. In this paper, we prove that the Zsigmondy set defined by iteration of weighted homogeneous polynomial is a finite set. In other words, let $f_t(x)$ be a weighted homogeneous polynomial of degree $d$ and weight $e.$ Let $t\in \mathbb{Q}\setminus \{0\}$ and let $Z(f_t, 0)$ be the Zsigmondy set for the zero orbit $\{f^n_t(0)\;\vert\;n\in\mathbb{N}\},$ then there exists a constant $k=k_{t, e, d}>0$ such that $\sharp Z(f_t,0) \leq k.$

Key Words: Primitive prime divisors, height function, weighted homogeneous polynomial.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 11G50.