This question is closely related to this :
The object is to find a polynomial of degree $5$ with non-negative integer coefficients for which
- $f(n)$ is composite for the integers $n$ with $0\le n\le 10^4$
- There is a non-negative integer $n$, such that $f(n)$ is prime
Let $M$ be the maximum of the coefficients of $f(x)$
Which is the smallest $M$ for which a polynomial $f(x)$ with the desired properties exist ?
$$29x^5 + x^4 + 24x^3 + 61x^2 + 60x + 210$$ is a polynomial for which the smallest prime value occurs at $n=10579$ , hence the smallest $M$ is at most $210$