Find the smallest odd integer $ k$ such that: for every $ 3-$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.
Quickly we can see that $ k\leq 7$.
Suppose $ f$ has a linear factor in $ \mathbb{Z}[n]$. Then
$ f(n) = (n-a)g(n)$
where $ g$ is a quadratic polynomial with integer coefficients. But $ |g(n)|=1$ is possible only for four values of $ n$ and $ |n-a|=1$ only for two values of $ n$. Hence $ f(n)$ can be prime at most for $ 6$ values of $ n$. Therefore if $ |f(n)|$ is prime for $ 7$ or more values of $ n$, then $ f$ is irreducible.
There is definitely better though. I'm willing to bet $ k=5$.
$ k>3$ because
$ f(n)=(n-3)(n^2+1)$ is prime for $ n=0,2,4$ and reducible.