lean_workbook_plus_15378

First, note that, the set of primes $\mathcal{P}=\{p:\exists n \in\mathbb{Z}^+,p\mid 3n^2+1\}$ is infinite (well known, via the fact that $-3$ is a quadratic residue, modulo prime $p\equiv 1\pmod{6}$). Now, for any such $p\in \mathcal{P}$, note, $p\mid 3(p-n)^2+1$ and $p\mid 3n^2+1$. Let $k=\min\{n,p-n\}$. Note that, $p>2k$, and $p\mid 3k^2+1$, as desired.