Exercise 13, § 1

Let f be a nonconstant polynomial in one indeterminate over Z. Show that for each integer k ≥ 1 there exist infinitely many integers n \in N such that f(n) is nonzero and divisible by at least k distinct primes (one can proceed by induction on k : if n \in N is such that f(n) is nonzero and divisible by at least k distinct primes, consider the polynomial f(n + Xf(n)')).