Prime Factors of Dynamical Sequences

2009/03/07 by Faber, Xander, Granville, Andrew
#11B37 (Primary) #11D59 (Secondary) #37F10 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0903.1344

Abstract

Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x₀, 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.

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