Reduction of the Wagstaff Chebyshev sufficiency conjecture to the Pell Primitive Pair Conjecture

LaTeX source, reproducibility scripts, and certified data for the paper "Reduction of the Wagstaff Chebyshev sufficiency conjecture to the Pell Primitive Pair Conjecture" by Alexey Dolotov (v3.6).

Abstract

For W_p = (2^p + 1)/3 and omega_3 = 3 + 2*sqrt(2), the congruence omega_3^{(W_p+1)/2} = -1 (mod W_p) (Condition II) holds for every prime W_p; whether it implies primality is the open Wagstaff Chebyshev sufficiency conjecture — equivalently the open "only if" direction of the Vrba–Reix Wagstaff primality test (S_{i+1} = S_i^2 - 2; open since 2008, standing 500-euro reward). This paper reduces the conjecture to a quantitative residual: the Pell Primitive Pair Conjecture (PPPC), a general-d nonexistence-of-compatible-triples (NCT) statement on the split branch, and a mild Wieferich–Wagstaff exclusion W = 0 (unconditional for all factors below 2^64). On the way it proves, unconditionally: NCT for every odd d <= 200 and at fifteen further parity-unblocked values in (200, 400]; the Order-Pinning, Multi-Factor Pinning, and Second-Moment reductions; a Pair Separation Theorem; quartic residuosity of 2 at primitive Pell divisors; and an exact-AP characterization. The inert branch collapses, via these reductions and one computational input (the Platinum Lemma, a 684,965,381-row enumeration), to PPPC. The reduction is conditional, not an unconditional proof.

This is the companion to the Brillhart–Lehmer–Selfridge primality-proofs paper (see Cite this work).

Layout

.
├── paper/
│   ├── wagstaff_chebyshev_reduction_v3.6.tex   LaTeX source (62 pp)
│   ├── wagstaff_chebyshev_reduction_v3.6.pdf
│   └── Makefile                                pdflatex targets
├── scripts/                                    reproducibility scripts
│   ├── README.md                               script -> paper section map
│   │   NCT fixed-d certificates (cor:nct-fixed-d-certificate):
│   │     cp351/cp353/cp356/cp358_nct_certificate*.py  per-d closures
│   │     cp356_aprcl_recert.py, cp358_aprcl_range.py  APR-CL re-cert
│   │     cp365_nct_certificate_bundle.py        consolidated static artifact
│   │   Inert/cross-case (Platinum, Second-Moment, pinning, exact-AP):
│   │     platinum_lemma.py, second_moment_reduction.py,
│   │     multi_factor_pinning.py, exact_ap_density.py, nct_*.py
│   │   Corner / falsification censuses:
│   │     cp354_diag_corner_gcd.py, cp358_corner_sweep_d1000.py,
│   │     cp358_falsification_prewindow.py
│   │   Vrba–Reix equivalence:  cp361_vrba_reix_check.py
│   │   Independent verification (preliminaries + inert foundations):
│   │     cp362_verify_preliminaries.py, cp363_verify_inert_foundations.py
│   │   Survey pipeline:  survey.py, build_clean.py, audit.py, verify_sample.py
├── data/
│   ├── nct_certificates.json   static fixed-d certificates: pinned Psi_{4d}
│   │                           factorizations + APR-CL + dispositions for all
│   │                           24 closures in (100,400] (FactorDB-independent)
│   ├── danger_triple_data.json V_{114}, V_{134}, V_{662} factorisations
│   ├── sample_1000.csv         first 1000 rows of the survey CSV
│   ├── SHA256SUMS              hashes
│   └── README.md               data dictionary
├── reproducibility.md          end-to-end reproduction walkthrough
├── CITATION.cff   .zenodo.json   LICENSE   README.md

Cite this work

• Paper (arXiv): pending submission — arXiv ID will be inserted here
• This bundle (Zenodo): DOI minted on release
• Inert-factor survey CSV (Zenodo): 10.5281/zenodo.19496206 (802 MB, 15,587,021 rows — the Platinum enumeration data)
• Companion BLS primality-proofs paper (Zenodo): 10.5281/zenodo.19645478

A machine-readable citation is in CITATION.cff.

Reproduce the central claims

NCT closures (no FactorDB needed). data/nct_certificates.json pins, for every fixed-d closure in (100, 400], the complete factorization of the Pell primitive part Psi_{4d} with each factor APR-CL-certified and its certificate disposition. Re-generate / re-verify it with python3 scripts/cp365_nct_certificate_bundle.py (PARI/GP gp required for APR-CL; the embedded local factorizations are checked by exact product identity).

Vrba–Reix equivalence. python3 scripts/cp361_vrba_reix_check.py confirms the recurrence test matches Condition (II) and tracks primality of W_p.

Survey CSV (Platinum enumeration). Download the 802 MB CSV from Zenodo (10.5281/zenodo.19496206), verify shasum -a 256 -c data/SHA256SUMS, audit with python3 scripts/audit.py, spot-check with python3 scripts/verify_sample.py data/sample_1000.csv.

See reproducibility.md for the full procedure.

License

MIT (code) / CC-BY-4.0 (paper, data). See LICENSE.