Research log: Landau 3 / Legendre

Every numerical claim below was taken from a paper, data page, or local replay. Community gap tables are identified as community evidence, not substituted for the published Sorenson--Webster computation.

2026-08-27: statement and analytic context

János Pintz, Landau's problems on primes, Journal de Théorie des Nombres de Bordeaux 21 (2009), 357--404. This is the survey used to identify the four 1912 Landau problems and their historical status. The Numdam PDF was also opened.
R. C. Baker, G. Harman, and J. Pintz, The Difference Between Consecutive Primes, II, Proceedings of the London Mathematical Society 83 (2001), 532--562. They prove that $[x-x^{0.525},x]$ contains a prime for all sufficiently large $x$ . Their constant is effective in principle but not made explicit.
Marc Chamberland and Armin Straub, Weakening the Legendre Conjecture, arXiv:2602.22502v1 (2026); the full HTML was replayed line by line. Under RH, Theorem 3.1 uses the Carneiro--Milinovich--Soundararajan interval $[y,y+(22/25)\sqrt y\log y]$ . Their sufficient inequality (5) is $x^{\delta/2}\geq(22/25)\log x$ . Combining it with the finite computation below gives the printed threshold $\delta=0.2253$ .
Thomas Campbell, On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares, arXiv:2603.10356v2 (2026); the full HTML was inspected. It proves an $\Omega\leq3$ result for every square interval. Its finite bridge uses the Sorenson--Webster range and an explicit prime-gap bound. It does not prove a prime exists between every pair of squares.
Chamberland--Straub record the current explicit consecutive-cubes threshold $n\geq\exp(\exp(32.537))$ , due to Cully-Hugill, and explain the route from Ingham's 1937 prime-gap work. The older explicit paper arXiv:0810.2113 was found during the same lookup.

The search for a later paper citing or improving the exact decimal $0.2253$ found no successor. That absence is only a search result, not a priority claim.

2026-08-27: computational record

Jonathan Sorenson and Jonathan Webster, An algorithm and computation to verify Legendre's conjecture up to $7\cdot10^{13}$ , Research in Number Theory 11, article 4, DOI 10.1007/s40993-024-00589-4. The journal version verifies the stronger Oppermann conjecture through $N=7.05\cdot10^{13}$ , exceeding $2^{46}$ and the prior $2\cdot10^9$ range. Correctness is unconditional; Cramer's model is used only in runtime analysis.
arXiv:2401.13753 is the earlier preprint, whose v1 abstract reports $3.33\cdot10^{13}$ . The journal version, not that stale abstract, is the record used here.
The authors' OLC source and data repository was cloned with full history at commit 5cdaa95f0a4b1428a05480cc1c69d556a8f9517a. Its checked-in logs are audited separately. A partial public upload cannot be used to revise the paper's stated result.

Exact replay of the headline arithmetic:

70{,}500{,}000{,}000{,}000^2 =4{,}970{,}250{,}000{,}000{,}000{,}000{,}000{,}000{,}000.

2026-08-27: prime-gap tables

Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, official paper PDF, Mathematics of Computation 83 (2014), 2033--2060, DOI 10.1090/S0025-5718-2013-02787-1. They computed all gaps below $4\cdot10^{18}$ . Their largest gap in that range is $1476$ , following $1425172824437699411$ , and they checked $g(p)<\log^2 p$ for $11\leq p\leq4\cdot10^{18}$ .
The corresponding author-maintained gap data page and Goldbach computation page were opened. The page states the computed and independently double-checked scopes separately.
Thomas Nicely and Bertil Nyman, New Prime Gaps Between $10^{15}$ and $5\cdot10^{16}$ , Journal of Integer Sequences 6 (2003), article 03.3.1, was opened as an earlier exhaustive computation.
The Prime Gap List community coverage page reports successive exhaustive limits $2^{64}$ in 2018, $2^{65}$ in 2025, $2^{66}$ in February 2026, and $10^{20}$ on 8 May 2026. Its data repository was opened. These later endpoints are community records linked to forum announcements, not a replacement for a peer-reviewed computation.

2026-08-27: OEIS replay

OEIS A014085 is $a(n)=\pi((n+1)^2)-\pi(n^2)$ . Its b-file contains 10,001 terms and cites Sorenson--Webster.
OEIS A005250 lists record prime-gap sizes.
OEIS A002386 lists the lower primes starting those record gaps.

The repository helper scripts/oeis_lookup.py was run on all three sequence IDs. It recovered the same definitions and references.

Failed or incomplete lookups

scripts/arxiv_fetch.py downloaded arXiv PDFs 2401.13753, 2602.22502, and 2603.10356 to temporary files, but could not extract text because pdftotext is absent. The arXiv abstracts and full HTML versions were then inspected directly.
The public OLC repository advertises the $7.05\cdot10^{13}$ result, but the checked-in detailed logs stop earlier. The pinned audit below records the exact scope. This is an artifact-availability result, not evidence against the paper.

2026-08-27: verified artifacts

The exact overlap equation has numerical root $0.22524401991936525179\ldots$ . Rational interval arithmetic certifies the clean value $901/4000=0.22525$ , with an overlap lower margin greater than $0.0000797000675$ . This is below the $0.2253$ printed at the end of Chamberland--Straub Section 3.
The OLC audit pins commit 5cdaa95f0a4b1428a05480cc1c69d556a8f9517a and selects 2,560 tracked worker *.out blobs. It finds 322,073 cumulative rows, 318,840 distinct intervals, 3,233 duplicate interval rows, no invariant failures, five union components, and four holes. The public maximum endpoint is $31{,}894{,}400{,}000{,}352$ .
The edge certificate covers every $n\in[2^{32}-100000,2^{32}-1]$ . Its square-height range is $[18445885090250351616,2^{64}]$ . Both strict Oppermann halves have a least prime witness for all 100,000 values.
The reported Oliveira e Silva--Herzog--Pardi gap row was independently replayed. The numbers $1425172824437699411$ and $1425172824437700887$ are prime, and every odd integer strictly between them is composite, so the gap is $1476$ . This verifies the row, not the exhaustiveness of the full $4\cdot10^{18}$ table.
Jonathan Sorenson and Jonathan Webster, Strong Pseudoprimes to Twelve Prime Bases, Mathematics of Computation 86 (2017), prove $\psi_{12}=318665857834031151167461>2^{64}$ . Their paper PDF was opened. This makes the first-twelve-prime-base Miller--Rabin verifier deterministic throughout the edge certificate.

Replay commands:

problems/landau-legendre/compute/q1/run_all.sh
problems/landau-legendre/compute/q1/replay_olc.sh /path/to/olc

Chamberland and Straub, Weakening the Legendre Conjecture, arXiv:2602.22502v1, was refetched with scripts/arxiv_fetch.py. The arXiv metadata still list v1, dated 26 February 2026. The full HTML, Section 3 was replayed at inequalities (5) and (6). The proof states that (5) is sufficient for (6), while (6) is the actual comparison needed to invoke the RH short-interval theorem.
Searches for the strings 0.22525 and 0.225244 together with the paper title found no later relevant arXiv result. This is a search result, not a priority claim.
The q3 exact certificate verifies $d=4504880398387/20000000000000=0.22524401991935$ . Its upper bound for the old condition (5) log margin is negative, below $-2.0321572\cdot10^{-13}$ . Retaining the positive quadratic Taylor term from (6) gives a certified log margin above $1.5199\cdot10^{-14}$ and a derivative lower bound above $1.9355$ .
A separate C long double implementation obtains the corresponding margins $-2.0328270\cdot10^{-13}$ , $1.5617425\cdot10^{-14}$ , and $1.9603180$ . The exact rational verifier is authoritative.

Replay command:

problems/landau-legendre/compute/q3/run_all.sh