  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.
