Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers

Abstract

The Wagstaff numbers $W_p = (2^p + 1)/3$ for odd primes $p$ are the natural $+1$ companions of the Mersenne numbers. Known primality proofs for $W_p$ with $p \geq 2617$ rely on the elliptic-curve primality proving algorithm of Atkin-Morain; Chebyshev/Lucas-type tests, while available as compositeness criteria, remain conjectural on the sufficiency side. We present fully verified primality proofs of $W_{2617}$ (788 digits), $W_{10501}$ (3161 digits), and $W_{12391}$ (3730 digits), independent of ECPP and relying only on classical $N-1$ machinery. The proofs apply the Brillhart-Lehmer-Selfridge (BLS) $N-1$ criterion to the cyclotomic decomposition $2^{p-1} - 1 = \prod_{d \mid p-1} Φ_d(2)$, harvesting factored content from the Cunningham project tables (used as evidence) and FactorDB (used only as a discovery aid, with every retrieved factor re-certified). As an independent check on the $\mathbb{Z}[\sqrt{2}]$ arithmetic implementation, the Chua $N+1$ congruence $ω_3^{(W_p + 1)/2} \equiv -1 \pmod{W_p}$ -- the $a=3$ case of the Chua framework with $ω_3 = 3 + 2\sqrt{2}$, a necessary condition for Wagstaff primality -- is verified at each $W_p$. BLS $N-1$ requires $p-1$ sufficiently smooth that enough cyclotomic factors $Φ_d(2)$ are fully factored. On the factorisation data consulted (Cunningham project tables and FactorDB, January-April 2026), $p = 10501$ and $p = 12391$ are the only exponents above $2617$ in the known Wagstaff prime/probable-prime list meeting this ceiling. Every cofactor primality is certified unconditionally by APR-CL; the method is independent of the Chebyshev sufficiency conjecture, and every step is reproducible from the archived scripts.