Abstract:We study the parametric subfamily p = 3m(m+1) + 1 with m = 2^a 3^b - 1, a,b in 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 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.
From: Hassane Bakkaoui M [view email]
[v1]
Wed, 17 Jun 2026 09:41:36 UTC (16 KB)
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