Prime factors of dynamical sequences

Abstract

Let fðtÞ A QðtÞ have degree d f 2. For a given rational number x0, define xnþ1 ¼ fðxnÞ for each nf 0. If this sequence is not eventually periodic, and if f does not lie in one of two explicitly determined a‰ne conjugacy classes of rational functions, then xnþ1 � xn has a primitive prime factor in its numerator for all su‰ciently large n. The same result holds for the exceptional maps provided that one looks for primitive prime fac- tors in the denominator of xnþ1 � xn. Hence the result for each rational function f of degree at least 2 implies (a new proof) that there are infinitely many primes. The question of primi- tive prime factors of xnþDxn is also discussed for D uniformly bounded.