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arxiv: 2605.18555 · v1 · pith:L4U7DN7Enew · submitted 2026-05-18 · 🧮 math.NT

Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers

Alexey Dolotov This is my paper

Pith reviewed 2026-05-20 08:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords Wagstaff numbersBrillhart-Lehmer-Selfridge criterioncyclotomic factorizationN-1 primality testAPR-CL certificatesCunningham tables
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The pith

Three large Wagstaff numbers receive unconditional primality proofs via classical N-1 factorization alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the Wagstaff numbers W_2617, W_10501 and W_12391 are prime by applying the Brillhart-Lehmer-Selfridge N-1 test to the complete factorization of 2^{p-1} - 1 into cyclotomic polynomials. Every cofactor is certified by APR-CL, and the entire chain is reproduced from archived data without using elliptic-curve methods. A reader cares because these proofs stand on ordinary integer arithmetic and publicly tabulated factors rather than on conjectural sufficiency conditions or external elliptic machinery.

Core claim

The central claim is that W_p for p = 2617, 10501 and 12391 are prime, established by verifying the BLS witness condition once the product of all Phi_d(2) dividing 2^{p-1} - 1 has been fully factored and each cofactor primality has been certified by APR-CL.

What carries the argument

The Brillhart-Lehmer-Selfridge N-1 criterion applied to the cyclotomic decomposition 2^{p-1} - 1 = product_{d | p-1} Phi_d(2), with all cofactors certified by APR-CL.

If this is right

  • The three numbers may now serve as bases for further deterministic number-theoretic constructions that require proven primes.
  • Any future Wagstaff probable prime whose exponent yields sufficiently smooth p-1 can be subjected to the same BLS treatment once its cyclotomic factors are certified.
  • The method supplies a reproducible template that separates the task of finding factors from the task of certifying primality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If Cunningham tables continue to grow, the same technique could certify additional Wagstaff numbers beyond the three treated here.
  • The independent Chua N+1 congruence check already performed in the paper could be turned into a routine cross-validation step for any future BLS proof on this family.

Load-bearing premise

The factorization records retrieved from the Cunningham project tables are correct and complete for every cyclotomic polynomial that appears.

What would settle it

A single composite cofactor among the Phi_d(2) factors that was treated as prime, or a failed verification of any APR-CL certificate.

read the original 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 $2^{p-1} - 1 = \prod_{d \mid p-1} \Phi_d(2)$, 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 $\omega_3^{(W_p + 1)/2} \equiv -1 \pmod{W_p}$ -- the $a=3$ case of the Chua framework with $\omega_3 = 3 + 2\sqrt{2}$, a necessary condition for Wagstaff primality -- is verified at each $W_p$. BLS $N-1$ requires $p-1$ sufficiently smooth that enough cyclotomic factors $\Phi_d(2)$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to deliver fully verified primality proofs for three Wagstaff numbers W_2617 (788 digits), W_10501 (3161 digits), and W_12391 (3730 digits) by applying the Brillhart-Lehmer-Selfridge N-1 criterion to the cyclotomic decomposition 2^{p-1}-1 = ∏_{d|p-1} Φ_d(2). Factorization data are taken from the Cunningham project tables (treated as evidence, with all factors re-certified) and FactorDB (discovery aid only); all cofactors are certified prime via APR-CL. An independent verification of the Chua N+1 congruence ω_3^{(W_p+1)/2} ≡ -1 mod W_p (with ω_3 = 3 + 2√2) is performed for each number as a check on the ℤ[√2] arithmetic implementation. The proofs are asserted to be deterministic, reproducible from archived scripts, and independent of ECPP.

Significance. If the reported cyclotomic factorizations and APR-CL certificates hold, the work supplies classical N-1 primality proofs for three of the largest known Wagstaff primes, a notable advance given that proofs for p ≥ 2617 have previously depended on elliptic-curve methods. The explicit re-certification of external factors, use of machine-checkable APR-CL, and provision of reproducible scripts constitute concrete strengths that enhance verifiability within the field of computational number theory.

major comments (2)
  1. [BLS N-1 application] BLS N-1 application (description of proofs for W_10501 and W_12391): the manuscript must state explicitly, for each W_p, the precise factored divisor F of N-1 (including its size relative to √N) and confirm that the product ∏_{q|F} (q-1) together with the chosen witnesses satisfies the full BLS threshold condition without gaps. The current reliance on the statement that “p = 10501 and p = 12391 are the only exponents meeting this ceiling” leaves the load-bearing arithmetic verification incomplete.
  2. [Cyclotomic factorization] Cyclotomic factorization section: while the paper notes that factorization data for the relevant Φ_d(2) are harvested from the Cunningham tables and re-certified, it should include a compact table or explicit list of the prime factors actually used for each p, together with the corresponding APR-CL certificate identifiers or verification commands. Without this, independent confirmation that F is complete and that no undetected composite cofactor remains is not possible from the manuscript alone.
minor comments (2)
  1. [Abstract] Abstract: the consultation window “January-April 2026” should be clarified (e.g., by reference to a specific archived snapshot of the Cunningham tables) to avoid any ambiguity about the exact data version employed.
  2. [Notation] Notation: ensure uniform use of W_p versus the expanded form (2^p + 1)/3 throughout; the abstract mixes both without explicit cross-reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight opportunities to strengthen the explicit verifiability of the BLS proofs, and we address each point below with planned revisions to the manuscript.

read point-by-point responses

  1. Referee: [BLS N-1 application] BLS N-1 application (description of proofs for W_10501 and W_12391): the manuscript must state explicitly, for each W_p, the precise factored divisor F of N-1 (including its size relative to √N) and confirm that the product ∏_{q|F} (q-1) together with the chosen witnesses satisfies the full BLS threshold condition without gaps. The current reliance on the statement that “p = 10501 and p = 12391 are the only exponents meeting this ceiling” leaves the load-bearing arithmetic verification incomplete.

    Authors: We agree that the manuscript should contain the explicit arithmetic details required to verify the BLS threshold without external computation. In the revision we will state, for each of W_10501 and W_12391, the precise value of the fully factored divisor F of N-1, its bit length relative to √W_p, the complete list of prime factors q dividing F, and the explicit confirmation that ∏(q-1) together with the square of the product of the chosen witnesses exceeds W_p. These quantities will be taken directly from the archived verification scripts. revision: yes

  2. Referee: [Cyclotomic factorization] Cyclotomic factorization section: while the paper notes that factorization data for the relevant Φ_d(2) are harvested from the Cunningham tables and re-certified, it should include a compact table or explicit list of the prime factors actually used for each p, together with the corresponding APR-CL certificate identifiers or verification commands. Without this, independent confirmation that F is complete and that no undetected composite cofactor remains is not possible from the manuscript alone.

    Authors: We accept that a compact table of the factors and their certificates will improve independent reproducibility. The revised manuscript will contain, for each p, a compact table listing the prime factors of the relevant Φ_d(2) that enter F, together with the APR-CL certificate identifiers (or the precise verification commands) for every cofactor. This addition will allow direct confirmation that F is complete and that all cofactors are prime. revision: yes

Circularity Check

0 steps flagged

No circularity: BLS proofs use external factorizations and standard criteria without self-referential reduction

full rationale

The paper applies the Brillhart-Lehmer-Selfridge N-1 criterion to Wagstaff numbers via the cyclotomic decomposition of 2^{p-1}-1, sourcing complete factorizations from Cunningham tables (treated explicitly as external evidence) and re-certifying all factors plus cofactors via independent APR-CL. The Chua N+1 congruence is presented only as a necessary-condition verification and independent implementation check. No equation, definition, or central claim reduces by construction to a fitted parameter, self-citation chain, or renamed input; the derivation remains self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard number-theoretic facts about cyclotomic polynomials and the BLS criterion; no free parameters are introduced and no new entities are postulated.

axioms (1)
  • standard math The Brillhart-Lehmer-Selfridge N-1 primality criterion and the factorization 2^{p-1}-1 = prod_{d|p-1} Phi_d(2) hold as stated in standard algebraic number theory.
    Invoked when the BLS test is applied to the cyclotomic decomposition of each Wagstaff number.

pith-pipeline@v0.9.0 · 5954 in / 1407 out tokens · 36558 ms · 2026-05-20T08:12:44.721708+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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