Prime factors of dynamical sequences

Let ( t ) ( t ) have degree d 2. For a given rational number x 0 , define x n 1 ( x n ) for each n 0. If this sequence is not eventually periodic, and if does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then x n 1 x n has a primitive prime factor in its numerator for all sufficiently large n . The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of x n 1 x n . Hence the result for each rational function of degree at least 2 implies (a new proof) that there are infinitely many primes. The question of primitive prime factors of x n x n is also discussed for uniformly bounded.