# Polynomial with non-negative “small” coefficients producing a prime “late”

This question is closely related to this :

How can I construct polynomials with "small" coefficients generating a prime "late"?

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 $210$

• How did you come by this particular metric for "smallest"? – Brian Tung Apr 3 '18 at 20:49
• I search for "hard cases" for the bunyakovsky-conjecture. And a "natural" bound for the smallest $n$ for which $f(n)$ is prime is a bound depending on the absolute values of the coefficients. Therefore the metric. – Peter Apr 3 '18 at 20:51