I'm not sure if this works but:1. Does there exist a non-constant polynomial P(x) with integer coefficients such that P(k) is prime for every positive integer k?
Justify your response.
Suppose there exists such a polynomial. P(0) =/= 0, as otherwise the x | P(x). Suppose P(0) = p for some prime number p. Then P(p) = ap^n + bp^n-1 + ... + p which is divisible by p, and is such a composite number. (1 is not a prime number, right?)