When is $n^2+n+p$ prime?

Question

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Behaviour of Polynomials in a PID!

Prove: if $p$ is a prime, and if $n^2+n+p$ is prime for $0\leq n \leq \sqrt{p/3}$, then it is also prime for $0 \leq n \leq p-2$.

This appeared on reddit recently, but no proof was posted. With $p=41$, it is Euler's famous prime-generating polynomial.

Answer 1

This follows by employing in Rabinowitsch's proof a Gauss bound, e.g. see Theorem 9.1 here.