Which Cubics are Irreducible?
For how many positive integers \(1 \leq k \leq 1000\) is the polynomial \(f_k(x)=x^3+x+k\) irreducible?
Details and assumptions
A polynomial with integer coefficients is called reducible if it equals to a product of two non-constant polynomials with integer coefficients. Otherwise, it is called irreducible.