[math-fun] Solving polynomial diophantine equations

Let P be a polynomial in one variable with integer coefficients. Suppose that (*) For all n in Z+, P(k) == 0 (mod) n has a solution in some integer k = k(n). 1) Then does it follow that P(k) = 0 has a solution for some integer k ??? (If not, what is a simple counterexample?) 2) Is (*) equivalent to saying P(k) == 0 mod q has an integer solution k(q) for all prime powers q ? 3) What is known about the questions analogous to 1) and 2) for integer polynomials in 2 integer variables??? More than 2 integer variables? It's known that the equation 1^2 + . . . + n^2 = m^2 has only one non-trivial solution: (n,m) = (24,70). 4) What solutions (n,m) exist if the exponent 2 is replaced everywhere by an integer p > 2 ??? --Dan