Openai 2026 prime predecessors even number prime factors

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lemma_3_1: Any fixed log-power saving in the sum of chi(p)p^(-1+it) over primes in an interval inside [N,2N], x^tau/2 <= N <= x^eta, for characters of modulus at most a power of log x and (log x)^B0 <= |t| <= x^2; the release's Artin-conjecture manuscript cites it. Claims checked only.

lemma_3_3: Sums of n^(iv) over an arithmetic progression of modulus at most a power of log x in an interval inside [N',2N'], exp(log x/(log log x)^2) <= N' <= 2x^5, save exp(-log x/(log log x)^C3) for heights exp(log x/(2(log log x)^2)) <= |v| <= 4x^3; it feeds Lemma 3.1, and the release's Artin-conjecture manuscript cites it. Claims checked only.

theorem_1_1: The manuscript's main claim: infinitely many primes p for which p-1 is squarefree with an even number of prime factors, that is with mu(p-1)=1. Claims checked only.

OpenAI, Prime Predecessors with an Even Number of Prime Factors, OpenAI Math Release preprint, September 17, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_prime_predecessors_even_number_prime_factors.pdf, and the release's TeX bundle in that folder (build/) is the TeX source cited on this card. The license term was read in the release's root LICENSE file at that revision on 2026-10-12.

@misc{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026,
  author = {{OpenAI}},
  title = {{Prime Predecessors with an Even Number of Prime Factors}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf}{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}},
  year = {2026}
}

The held PDF has 45 pages with a text layer; its title page is dated September 17, 2026 and its metadata records a creation date of 17 September 2026. The TeX source is build/main.tex (141 lines: preamble, abstract, Theorem 1.1 and the introduction), which inputs seven section files build/sections/01-preliminaries.tex to 07-extraction.tex (3,263 lines together; they print as Sections 2--8) and build/references.tex (39 lines), 3,443 lines in all.

Attestation. The release README says that the collection's manuscripts and supporting proof artifacts were "produced by an internal OpenAI model", that "Some outputs build upon earlier results produced by the models", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says that "The vast majority of results were obtained with the same procedure" using an unreleased model, and names as exceptions work on a zero-free region for the Riemann zeta function and on the Hodge conjecture for CM abelian varieties, neither of which is this family's subject. The manuscript's own README carries only the title, the author line "OpenAI", the date September 17, 2026 and the citation block, and adds no sentence about authorship or assistance; the title page names OpenAI as the sole author, and the text says nothing about how it was produced. These are the source's own statements, recorded as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here, and nothing on this card is independently reviewed.

Formalization. The release lists no formalization of this manuscript: its Lean catalog lean/formalization.yaml has no entry for it at this revision, and the release has no Lean page for its family.

Companions. The release files this manuscript in a family titled "Prime-factor statistics of $p-1$" in its manuscript map and "The Ford--Konyagin--Luca conjecture on prime predecessors" in its overview, whose summary describes only the other two members. Weighted dilation graphs, smooth shifted primes and totient fibers (September 24, 2026), filed at the weighted-dilation card, is the paper this manuscript calls S: it imports from S, without proof, the stated forms of its classical estimates (Section 2), its transference and residual ideal operator theorems and integer bilinear estimate (Section 4), and its block sieve and proxy estimates (Section 8), and says its own new work is to put a prime in the long factor slot where S's shifted-correlation argument has a rough integer, and to keep the same parity sign at both ends of the correlation. The bibliography's link to S names S's release folder, dated a week after this manuscript; S does not cite this manuscript, as its card records. The Poisson--Dirichlet law for prime predecessors, dated September 24, 2026 and filed at the Poisson--Dirichlet card, is the third member; neither manuscript cites the other. That manuscript states the long prime polynomial and logarithmic-phase estimates, in the same ranges as Lemmas 3.1 and 3.3 here, as its Lemmas 2.3 and 2.4, and proves them in its own Appendix A. The release holds no other dated version of this manuscript, and its history file records no change to it. Outside the family, the release's Artin-conjecture manuscript, Primitive roots for every admissible integer base (October 4, 2026), filed at the Artin card, lists this manuscript as its reference [22] with the note "Section 3, especially Lemmas 3.1 and 3.3". Its three citations all sit in the proof of its Proposition 10.3, a rough-factor Type II estimate on which its Theorem 1.1 rests. They take Lemma 3.1, Lemma 3.3 (for the range of a bound quoted from the Poisson--Dirichlet manuscript) and the classical Siegel--Walfisz estimate that Proposition 2.1 here records without proof, and nothing from Sections 4--8 or from Theorem 1.1; the lemma pages give the citing passages. The companions' arguments were not read for this card; only the statements of the Poisson--Dirichlet Lemmas 2.3 and 2.4 were compared with Lemmas 3.1 and 3.3, and the opening paragraph of its Appendix A was read. Of the Artin manuscript only the citing passages were read.

Read status: claims checked for Theorem 1.1, Lemma 3.1, Lemma 3.3 and Proposition 2.1, read clause by clause in the release's TeX source (build/main.tex, abstract lines 40--44 and label thm:main at lines 53--57; build/sections/01-preliminaries.tex, conventions lines 3--17 and label pre:primes at lines 24--40; build/sections/02-long-prime.tex, labels lp:prime-polynomial at lines 8--21 and lp:log-phase at lines 185--197); the statements of Theorems 4.1, 6.1 and 7.1 and of the lemmas numbered 3.2, 3.4--3.6, 5.1--5.3 and 8.1--8.9 were read as statements; all proofs were read for their structure only and no step was checked; nothing here is independently reviewed. The citing passages of the Artin manuscript were read in its TeX source at the same revision (sections/08-type-ii.tex lines 321--536, sections/12-construction.tex lines 83--127, where Lemma 12.1 invokes the proof of Proposition 10.3, and references.tex lines 148--153). No search was made for earlier results on the parity of $\Omega(p-1)$.

Contents

The PDF has 45 pages: eight sections and five references. Results share one counter numbered within sections, and the printed page numbers are the PDF's.

• Section 1, Introduction (pp. 1--2). Defines $\Omega$ and $\omega$, states Theorem 1.1 (p. 1), and says that squarefreeness makes the two counting conventions agree. The plan: write $p=2u+1$ and build nonnegative weights on odd $u$ with $\Omega(u)$ odd, with enough bilinear distribution to detect primes $2u+1$ and a negligible nonsquarefree part. It names the parity obstruction of sieve theory, cites the asymptotic sieve of Friedlander and Iwaniec (their Section 1) for the role of bilinear information and the smooth-shifted-prime work of Baker--Harman and Lichtman as context, and says the parity condition is not deduced from smoothness. Two analytic extensions are announced: cancellation in prime polynomials up to height $x^2$ through a logarithmic-phase estimate and a weak zero-free strip, and the cancellation of a shared dilation's sign, which occurs twice in a lifted correlation, by complete multiplicativity.
• Section 2, Conventions and classical estimates (pp. 2--4). Sets $L=\log x$, $T=\log L$, $W=\exp(\sqrt L)$; constants may depend on all previously fixed parameters and no effective threshold for $x$ is asserted. Proposition 2.1 (prime number theorem with any fixed log-power error, Mertens, Siegel--Walfisz for moduli up to $(\log y)^C$, constants not necessarily effective), Proposition 2.2 (the primitive-character large sieve, divisor moments, and mean values of Dirichlet polynomials on intervals and at spaced points), Proposition 2.3 (first and second derivative tests), all recorded without proof in the forms of S's Section 2 (Proposition 2.2 is its Theorem 2.3 and Lemmas 2.5--2.6, Proposition 2.3 its Lemma 2.7), and Lemma 2.4 (smooth separation of variables by Fourier inversion), an elementary version of S's Lemma 2.8 proved here.
• Section 3, A long prime polynomial (pp. 4--11). States Lemma 3.1 (p. 4). Lemma 3.2 (p. 5): a Vinogradov mean-value bound uniform in the degree, $J_{s,k}(M)\le\exp(C_1k^{C_2})M^{2s-k(k+1)/2+1/100}$ for some $s\le C_1k^4$, by Linnik's $p$-adic iteration (exposition cited to Wooley 2012, Section 2; efficient congruencing not used). Lemma 3.3 (p. 7). Lemmas 3.4--3.6 (pp. 8--10), for Dirichlet $L$-functions $D(s,\chi)$ of modulus at most $L^C$: $\log|D|\ll T^2$ within $10T^4/L$ of the line $\operatorname{Re}s=1$ at heights $1/2$ to $3x^3$, a local formula for $D'/D$ through $O(T^2)$ nearby zeros, and no zero in $\operatorname{Re}s\ge1-cT^2/L$, $1\le|\operatorname{Im}s|\le x^3$. The proof of Lemma 3.1 closes the section (pp. 10--11).
• Section 4, A prime-slot shifted correlation with a common parity twist (pp. 12--22). Fixes disjoint groups of primes (small ones in $[\exp(L^a),\exp(L^b)]$, big ones in $[\exp(L^c),\exp(L^d)]$, $0<a<b<c<d<0.47$, of order $T$ groups of each kind), marked weights $W_{\mathbf t}$ that count marked group-prime divisors with a damping factor $q_0$ per unmarked one, a sign $\varepsilon$ equal to $1$ or to $-1$ raised to the number of group-prime factors counted with multiplicity, and endpoint cores $F(n)$: sums over the factorizations $n_*=mpr$, where $n_*$ is the part of $n$ prime to the groups, of a $W$-rough coefficient $\alpha_m$ twisted by a character and a phase, a prime $p$ in a slot $I_p\subset[x^\tau,x^\eta]$, $\eta<1/4$, with a phase, and a coefficient $c_r$, with $|\alpha_m|,|c_r|\le L^C$. Theorem 4.1 (p. 13): if the coefficients $\alpha$ of one core have discrepancy at most $L^{-B}$ against characters of modulus up to $L^B$ and phases $m^{it}$, $|t|\le L^B$, then the harmonic shifted correlation of $\overline{F(z)}$ with $G(z+k)$, weighted by $W_\ell$ and the same sign at both ends, is $\ll L^{-D_0}$ at scales $Z$ within a power of $L$ of $x$ and for shifts $0<|k|\le L^C$. Propositions 4.2 and 4.3 restate the imported graph inputs (S's Theorem 3.5, Corollary 3.11 and Lemma 3.4; S's Theorem 4.1 and Lemma 4.3), and S's integer bilinear estimate (its Lemma 4.2) is quoted. The proof removes the shared dilation by $\varepsilon(D_R)^2=1$ ((4.18), p. 16), then treats the comparison terms by minor and major arcs, the local Mellin energy of Lemma 4.4 (p. 19, from S with proof), a factorial bound on the exceptional times of a small-prime polynomial, Lemma 3.1 on the prime slot, and the discrepancy hypothesis at low heights. Remark 4.5 (p. 22) fixes the order of choices.
• Section 5, Candidate weights and their mass (pp. 22--26). Ordinary bands $Q_j$ ($\log p\in[s_j,\tfrac65s_j]$, $s_j=s'2^{-j}L$) each contribute exactly $r_0$ primes, $r_0$ a fixed even integer; groups $\mathcal P_g$ in $[L^{1/10},L^{1/5}]$ and $[L^{3/10},L^{2/5}]$ (in $\log p$) carry the marks with $q_0=1/2$, $\ell=1$. The weight $A(u)$ keeps an odd number of group-prime factors, so $A(u)>0$ forces $\Omega(u)$ odd and $\Omega(2u)$ even ((5.5), p. 23). Lemma 5.1 bounds the group part's tail; Lemma 5.2 (p. 24) gives the total mass $X_A\asymp(x/L)H_QH_P\gg xL^{-C_A}$; Lemma 5.3 (p. 25) shows that $u$ with $2u$ not squarefree carry $o(X_A/L)$.
• Section 6, Bilinear distribution (pp. 26--33). Theorem 6.1 (p. 26), the Type II estimate: for $mn=2u+1$ with $m,n$ in dyadic ranges inside $[x^{b_*},x^{1-b_*}]$, bounded-by-$L^C$ coefficients, and the $m$ coefficient $W$-rough and of small discrepancy, the sum against $A(u)\Psi(u/x)$ is $\ll xL^{-D_*}$, with no condition on the $n$ coefficient. Proof by a designated prime slot, a smooth allocation of the band primes, Fourier separation, Cauchy--Schwarz and a change of variables that turns the off-diagonal terms into correlations covered by Theorem 4.1, once for $\varepsilon=1$ and once for the parity sign. Remark 6.2 (p. 33) orders the later proxy precision after the discrepancy exponent.
• Section 7, Distribution in congruence classes (pp. 33--36). Theorem 7.1 (p. 33): the remainders $r(d)$ of the weight on $d\mid 2u+1$ against $X_A/\varphi(d)$, summed over odd squarefree $d\le x^\vartheta$, $\vartheta<1/2$, are $\ll xL^{-D}+X_AL^{-18}$, from one prime of the band $Q_0$, the large sieve for large conductors and Siegel--Walfisz for small ones.
• Section 8, Prime extraction (pp. 36--45). Imports Lemma 8.1 (block sieve, S's Lemma 2.9) and Lemma 8.2 (local-density proxies for the prime and $x^\gamma$-rough indicators on logarithmic cells of width $L^{-S_1}$, with their discrepancy and density bounds, S's Lemmas 7.3--7.4). Lemmas 8.3--8.4 (p. 39) give a positive pair bound by a masked sieve. Lemma 8.5 (p. 42) replaces tests by proxies through Theorem 6.1. Lemma 8.6 (p. 42) is the presieve to $P^-(2u+1)>x^{b_1}$, with the singular series $\mathfrak S=2\prod_{p\ge3}(1-(p-1)^{-2})$. Lemmas 8.7--8.8 (p. 43) remove the composites whose least prime factor lies in $(x^{b_1},x^{1/2-\kappa}]$ by Buchstab's identity, leaving at least $\mathfrak SX_A/(2L)$. Lemma 8.9 (p. 44) bounds the products of two primes near $\sqrt x$ by $C_{\rm bal}(\kappa+1/L)X_A/L$ up to $o(X_A/L)$. The proof of Theorem 1.1 (pp. 44--45) combines them.
• References (p. 45): Baker--Harman (1998), Friedlander--Iwaniec (1998), Lichtman (arXiv:2211.09641v1), S, and Wooley (2012).

The manuscript flags nothing as numerical, computer-assisted or conditional; it asserts no effective threshold for $x$, and its Siegel--Walfisz inputs may have ineffective constants. The release folder holds only the README, the PDF and the TeX build directory; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. A search of the problem pages' statements and assessments for the squarefreeness or the parity of the number of prime factors of $p-1$, for $\mu(p-1)$, $\Omega(p-1)$ and $\omega(p-1)$, and for shifted primes found no page whose question or recorded argument turns on these objects: the pages that use smooth shifted primes (Problems 821 and 1057) rest on the smoothness of $p-1$, which this manuscript does not address, and the mention of the parity problem on the page of Problem 385 concerns semiprimes in short intervals. Lemmas 3.1 and 3.3 reach problem pages only at second hand, through the Artin manuscript's Theorem 1.1, whose card records its own rows (Problems 985 and 429); this card adds none.