Unconditional Primality Certificates for the Hexagonal 3-smooth Family p = 3m(m+1) + 1: Deterministic Pocklington Witnesses and Arithmetic Filters

Abstract

We study the parametric subfamily $p = 3m(m+1) + 1$ with $m = 2^a 3^b - 1$, $a,b \in \mathbb{N}^*$, a 3-smooth slice of the centred hexagonal numbers $3m^2 + 3m + 1 = (m+1)^3 - m^3,$ from the point of view of unconditional primality certification via the Pocklington-Lehmer criterion. The 3-smoothness of $m+1 = 2^a 3^b$ yields, for every $(a,b)$, a fully factored divisor $F = 2^a 3^(b+1)$ of $p-1$ satisfying $F > \sqrt(p)$ unconditionally, reducing the certificate to two witnesses, for $q = 2$ and $q = 3$. Our main new contribution is a complete, deterministic characterisation of the two canonical witnesses. We prove that $w_2 = 5$ is a valid witness if and only if $a - b$ = 1, 2 (mod 4), by quadratic reciprocity; and that $w_3 = 7$ is a valid witness if and only if $m$ is not congruent to 2 (mod 7), by cubic reciprocity in $\mathbb{Z}[omega]$ using the explicit Eisenstein factorisation $p = ((1+m) - m \omega)((1+m) - m \omega^2)$. These two results turn the heuristic "5 and 7 always work" (which is in fact false) into exact congruence conditions, and yield a deterministic witness-selection rule. Alongside, three elementary arithmetic filters (mod 6, a (-3) quadratic-residue sieve, and a mod-7 forbidden-class test) remove about 87% of candidates at negligible cost. As a demonstration, a multi-core implementation produced four unconditional certificates on consumer hardware, the largest a prime of 29998 decimal digits.