Prime Factors of Dynamical Sequences

TL;DR

This paper proves that for certain rational functions, the differences in the generated sequences have primitive prime factors for large n, offering a new proof of the infinitude of primes related to these sequences.

Contribution

It introduces a novel proof that sequences from rational functions of degree at least 2 have infinitely many prime factors in their differences.

Findings

Differences x_{n+1}-x_n have primitive prime factors for large n.

Provides a new proof of the infinitude of primes for these sequences.

Applies to non-eventually periodic sequences from rational functions.

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.