Prime Factors of Dynamical Sequences

Abstract

Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference x_{n+1}-x_n has a primitive prime factor for all sufficiently large n. This result provides a new proof of the infinitude of primes for each rational function f of degree at least 2.

Corrected several typos and a non-critical error in Lemma 5. No change to the statements of the main theorems. To appear in Crelle's Journal