Research log — Brocard–Ramanujan factorial square

Every external lookup made during q2 is recorded here. Search snippets were used only to locate sources; mathematical claims in the attack rely on the linked paper itself, the checked computation, or the Lean proof.

2026-08-16 — q2

Sources read

• Abderrahim Makki Naciri, On the Brocard–Ramanujan Equation with 7-Free Integers and Prime Powers, INTEGERS 25 (2025), #A71, DOI/Zenodo record. The paper explicitly starts from $n!=(x-1)(x+1)$, and proves finiteness only under restrictions such as one neighbor being $k$-free or having few distinct prime divisors. It states that the unrestricted problem remains open. This was important scope control: q2's factorization is standard, while the full-prime-power Lean consequence and exact unitary-divisor experiment are quest artifacts, not a claimed literature novelty or an unrestricted solution.
• OEIS A034444, “number of unitary divisors.” It gives the definition $d\mid N$ with $\gcd(d,N/d)=1$, and equivalently says that each complete prime-power block of $N$ is either present or absent. This justifies the standard term unitary divisor and the count $2^{\omega(N)}$ before identifying complementary choices.
• Saša Novaković, The Diophantine equation $P(x)=\prod H_{n_i}$ (2026 preprint). Its introduction records that the integer Brocard–Ramanujan problem is open, cites Overholt's conditional result, and summarizes Naciri's restricted-neighbor result. It supplied a recent check that no unconditional factorization theorem was being overlooked.
• Berndt and Galway, On the Brocard–Ramanujan Diophantine equation $n!+1=m^2$, Ramanujan Journal 4 (2000), 41–42. The Illinois bibliographic record confirms that the paper reports a computation through $10^9$. q1 had already read the two-page paper and rerun only a labeled $10^7$ method slice; q2 did not reuse $10^9$ as its own bound.
• Brocard–Ramanujan problem for polynomials over finite fields, Finite Fields and Their Applications 110 (2026), 102731. The abstract concerns Carlitz factorials over $\mathbb F_q[T]$, not the integer equation; it was screened out after confirming that it does not supply the needed integer factorization obstruction.

Negative and failed lookups

• OEIS was searched for the initial q2 minimum-gap terms `1, 1, 11, 1, 97, 181` and for “unitary divisor factorial square Brocard.” No matching sequence was found. A365401, the first search hit, instead counts divisors of the largest square unitary divisor and is unrelated. Accordingly no OEIS or priority claim is made for the q2 gap data.
• Web/arXiv searches for `Brocard Ramanujan (m-1)(m+1) prime powers`, `consecutive coprime factors n!/4`, and the corresponding $p$-adic congruence found the sources above but no paper asserting an unrestricted conclusion from this exact prime-block split.
• The direct author-PDF URL faculty.math.illinois.edu/~berndt/articles/galway.pdf timed out in the q2 web fetch; the paper had already been read in q1 and its metadata was checked through the DOI and Illinois record instead.
• The normal HTML page for Erdős problem 398 returned HTTP 403 to the fetcher. Its text/LaTeX endpoint appeared in a later search and lists the Naciri paper; no theorem from that endpoint is used here.
• General searches also surfaced MathWorld/Wikipedia summaries and several nonstandard-host “solution” manuscripts. They were not used: their advertised conclusions conflict with the peer-reviewed sources' explicit open status, and q2 makes no novelty or solution claim.

2026-08-23 — q3

Sources opened

• OEIS A085692, fetched with scripts/oeis_lookup.py. It lists the square values $25,121,5041$ arising from the three known positive indices. This matches the folder statement; it does not assert that the list is complete.
• Crossref's record for Berndt–Galway, DOI 10.1023/A:1009873805276, was opened. It confirms the authors, title, journal, year, volume, and pages 41–42, but exposes no theorem text or abstract.
• The Springer article page and Springer PDF endpoint were opened but returned a browser challenge rather than the paper.

Failed lookup and scope consequence

• The Berndt author-PDF URL failed repeatedly with a TLS end-of-file error. Unpaywall and Semantic Scholar both report the article as closed and provide no repository copy. Therefore q3 does not treat the often-quoted $10^9$ as a replayed baseline or use it in any new claim. The attack below relies only on Wilson's theorem and exact arithmetic that is reproduced locally.
• The exact metadata checks were the Unpaywall DOI response and the Semantic Scholar DOI response. A CORE title/DOI search returned no copy, while the equivalent OpenAlex query was rate-limited.
• Further recovery attempts through the Internet Archive catalog, its capture index, the old UIUC author host, two alternate Illinois hosts (one, two), and an exact-title DuckDuckGo lookup produced no paper. A text-proxy attempt failed too. Encoded variants of the Springer PDF URL returned the same client-challenge HTML. These failures are recorded to distinguish a metadata check from a read of the theorem text.