Infinitely many primes congruent to one modulo six ($\#\{p\text{ prime}:p\equiv 1(\mathrm{mod}6)\}=\infty$)

Assume that $x^2\equiv -3(\mathrm{mod}p)$ with $p>3$ implies $p\equiv 1(\mathrm{mod}6)$. For a finite list of primes in that residue class, let $P$ be their product. Every prime divisor of $4P^2+3$ is greater than three and satisfies that quadratic modular congruence, hence has residue one modulo six. None is listed because $4P^2+3\equiv 3$ modulo a listed prime.

// Target: combinatorics.bigb