Can anyone help with number theory

Problem 5. (4 points.) Let f E Z[x] be a nonconstant polynomial. Prove that the set S = {p prime: there exist infinitely many positive integers n such that p | f (n)} is infinite. Problem 6. (4 points.) Let f E Z[x] be a nonconstant polynomial with the property that all the roots (in C) for the equation f (2) = 0 are distinct. Prove that there exist infinitely many positive integers n such that f(n) is not a perfect square. Problem 7. (1 points.) Let f(x) E Z[x] be a polynomial of degree 3. Prove that there exist infinitely many positive integers n such that f(n) is not a perfect square. Problem 8. (8 points.) Let f(x) = 23 + 23. Find all integers n with the property that f(n) is a perfect square