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

For any finite list of such prime numbers, let $P$ be their product. The integer $6P-1$ has no listed prime divisor, nor the divisors $2,3$. Its remaining prime factors have residues $1,-1$ modulo six, and at least one has residue $-1$ because their product has that residue. Thus no finite list is exhaustive.

// Target: number-theory.bigb