Openai 2026 primitive roots admissible integer base

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theorem_1_1: The claimed infinitude part of Artin's conjecture for every integer base other than -1 and the squares, with a lower bound of order x/(log x)^2 in each large dyadic interval; reduced to a sieve construction of primes with controlled predecessors and a uniform complete-splitting bound. Unverified.

theorem_1_2: The claimed analytic input: a zero-free strip of common width 10^{-6} for every finite-order Hecke L-function of every cyclotomic field containing the twelfth roots of unity, proved by comparing a reflected second moment of cubic theta coefficients with a Poisson evaluation. Unverified.

OpenAI, Primitive roots for every admissible integer base, OpenAI Math Release preprint, October 4, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026; the held PDF, primitive-roots-all-integer-bases.pdf in the release, is retained as openai_2026_primitive_roots_admissible_integer_base.pdf, and the release's TeX bundle sits in the same release folder.

@misc{OAI:Primitive-roots-for-every-admissible-integer-base-October-4-2026,
  author = {{OpenAI}},
  title = {{Primitive roots for every admissible integer base}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf}{OAI:Primitive-roots-for-every-admissible-integer-base-October-4-2026}},
  year = {2026}
}

Attestation as the release states it. The release's root README says the collection holds manuscripts "produced by an internal OpenAI model", that it "includes results at different stages of verification", that not all of them have Lean formalizations, and that "Some of the unformalized results could have issues". The manuscript's own README carries only the title, the author line "OpenAI", the date October 4, 2026 and the citation block above, and adds no sentence about human assistance or verification. The manuscript itself names no author beyond "OpenAI" and carries no statement on how it was produced. These are the source's historical attestations, not this corpus's review. No refereed publication, no arXiv version and no independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalog (lean/formalization.yaml) lists no formalization for this manuscript and has no page for its family.

Companions. The release groups this manuscript with Simultaneous primitive roots: a conditional lower bound for prime bases (same date), whose abstract claims that every fixed finite set of distinct positive primes is a set of simultaneous primitive roots for at least $cx/(\log x)^2$ primes in $(x,2x)$, conditionally on four stated analytic and sieve inputs taken from this manuscript; that companion is not held in this library. Three other release manuscripts enter as cited inputs and are held here: the marked Type II estimate is imported from the Poisson--Dirichlet law for prime predecessors (its Theorem 3.1), two elementary sieve lemmas are adapted with proofs from Weighted dilation graphs, smooth shifted primes and totient fibers (its Lemmas 2.9 and 7.4), and the analytic argument follows the reflection and additive-probe method of The Quasi-Riemann Hypothesis (its Part I, Sections 5--7) without importing its theorem. A fourth, Prime predecessors with an even number of prime factors, is cited in Section 10 for its Proposition 2.1 and Lemmas 3.1 and 3.3 and is not held here.

Read status: claims checked for Theorem 1.1, Theorem 1.2, Proposition 2.1, Proposition 2.2 and Lemmas 2.3--2.4, read clause by clause in the TeX source (sections/01-introduction.tex lines 17--45 and sections/01a-reduction.tex lines 18--38, 50--63, 69--76 and 104--113 of the TeX bundle; PDF pp. 2--6) on 2026-10-07; the proof of Theorem 1.1 (01a-reduction.tex lines 150--201, PDF p. 7) and Sections 3--12 and Appendix A were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The PDF has 92 pages; section numbers below are the printed ones, and the TeX bundle's section files are named in order.

• Section 1, Introduction (pp. 2--5; 01-introduction.tex). Defines the multiplicative order $\operatorname{ord}_p(a)$, a primitive root, and calls an integer $a$ admissible when $a\ne-1$ and $a$ is not a square (the two necessary conditions). States Theorem 1.1: for every admissible $a$ there are $c_a>0$ and $x_a\ge2$ such that for all real $x\ge x_a$ at least $c_a x/(\log x)^2$ primes $p\in(x,2x)$, $p\nmid a$, have $\operatorname{ord}_p(a)=p-1$; the predicted scale is $x/\log x$ and the constants may depend on $a$. States Theorem 1.2: for every cyclotomic field $F\supseteq\mu_{12}$ and every finite-order Hecke character $\eta$ of $F$, $L_F(s,\eta)$ has no zero in $\operatorname{Re}s>1-10^{-6}$, the principal character included with its pole allowed; the width is common to all such $F$ and $\eta$, while the constants in the proof may depend on them. Section 1.1 recalls Hooley's conditional proof of Artin's asymptotic under the Riemann hypothesis for the Kummer fields $\mathbb{Q}(\mu_n,a^{1/n})$, the unconditional results of Gupta--Murty (one of thirteen bases) and Heath-Brown (at most two prime bases fail, so one of $2,3,5$ works), the average results of Goldfeld, Stephens and Klurman--Shparlinski--Teräväinen, and the Thorner--Zaman simultaneous zero-free region with its possible exceptional zero. Sections 1.2--1.3 outline the two parts of the proof; Section 1.4 gives the plan and notation ($L=\log x$).
• Section 2, Reduction to controlled predecessors and complete splitting (pp. 5--8; 01a-reduction.tex). The index $i_p(a)=(p-1)/\operatorname{ord}_p(a)$; Hooley's index-divisor reduction. Proposition 2.1 (primes with controlled predecessors): for fixed $M\equiv0\pmod 8$, $c\in\{2,4\}$ and $u$ with $(u,M)=1$, $c\mid u-1$ and $((u-1)/c,M/c)=1$, there are $\gg_{M,c,u}x/L^2$ primes $p\in(x,2x)$ with $p\equiv u\pmod M$ and $p-1=crQ$, $Q>x^{0.9}$ prime, every prime factor of $r$ in $(\exp(L^{0.1}),\exp(L^{0.3}))$. Proposition 2.2 (uniform bound for complete splitting): for fixed $|a|>1$, $\delta_0=10^{-6}$, all large $x$ and all primes $q$ large in terms of $a$ with $q\le\exp(L^{0.3})$, the number of primes $p\in(x,2x]$ splitting completely in $K_q=\mathbb{Q}(\mu_q,a^{1/q})$ is $\ll_a x/(q(q-1)L)+x^{1-\delta_0}$, with constant and threshold independent of $q$ (for $a=2$ every $q\ge5$, with absolute constants). Lemma 2.3 (splitting test, proved in the manuscript): for $p\nmid aq$, $p$ splits completely in $K_q$ iff $p\equiv1\pmod q$ and $a^{(p-1)/q}\equiv1\pmod p$. Lemma 2.4 (proved in the manuscript): for admissible $a$ and $M=8\prod_{\ell\mid a,\ \ell\text{ odd}}\ell$ there are $c,u$ meeting Proposition 2.1's conditions with $(a/p)=-1$ for every prime $p\equiv u\pmod M$ (quadratic reciprocity and a seven-entry table for the squarefree kernels $-1,\pm2,\pm3,\pm6$). Proof of Theorem 1.1 (p. 7), summarized on the result page. Remark 2.5 on negative squares and odd perfect powers; the case $a=2$ can use $(M,c,u)=(8,4,5)$.
• Section 3, S-integers, residue symbols, and Gauss sums (pp. 8--15; 02-arithmetic.tex). For a fixed cyclotomic $F\supseteq\mu_{12}$ and a finite set $S$ of places (enlarged so that the ring of $S$-integers $R$ is principal and to include two excluded sets fixed later), chooses generators with fixed power classes, for which sextic reciprocity holds without a supplementary factor (Lemma 3.2), and identifies the two Hecke characters giving the cubic and sextic Gauss-sum phases (Lemmas 3.3--3.4). Inputs: Neukirch, Weil, Hoshi--Kanai (Davenport--Hasse).
• Section 4, The coefficient family and analytic comparison (pp. 15--18; 02a-coefficient-family.tex). Defines the Gaussian profile $P_G$, the coefficient sums $B_m(Z)$ over $\mathfrak{A}=\mathfrak{c}\mathfrak{n}^3$ with $\mathfrak{c}$ squarefree, the scales $a=9/10$, $b=133/1000$, $M_0=1033/1000$, and the row sets $\mathscr{B}_Z(\mathscr{K})$. States Proposition 4.1 (reflected second moment): $\sum|B_m(Z)|^2\ll Z^{\epsilon}P^{1/2}(D+D^2P/Z)$ over rows with powerful part of norm about $P$ and squarefree part about $D$, hence total mass $\ll Z^{2M_0-1+\epsilon}$. Then explains in prose how the comparison of a probe scalar $I_Z$ (upper bound $Z^{933/2000+\epsilon}$ from Proposition 4.1; evaluation $c_0f_\eta(Z)+O(Z^{7/15-.0004})$ from Poisson summation) gives $f_\eta(Z)\ll Z^{7/15-1/20000}$ and so excludes zeros of $L^S(s,\eta)$ in $\operatorname{Re}s>1-1/20000$. This is the outline the result page for Theorem 1.2 follows.
• Section 5, The normalized cubic theta input (pp. 18--27; 03-theta.tex). Proposition 5.1: a threefold central cover of $\mathrm{PGL}_2$ over $\mathbb{A}_F$ (the Kazhdan--Patterson $c=2$ cover modulo scalars) and a genuine automorphic representation $\Theta$ with one-dimensional local Whittaker spaces, with its exact lifts, exterior local values and complex local behavior; Corollary 5.2 identifies the exterior coefficient $\gamma_2(\mathfrak{c})/(q_{\mathfrak{c}}^{1/2}q_{\mathfrak{n}})$ of $B_m$. Lemmas on a finite torus cutoff, norm-sheet profiles and exact Gaussian smoothing. Appendix A supplies the normalization.
• Section 6, Reflection and the quadratic large sieve (pp. 27--41; 04-reflection.tex). Proves Proposition 4.1: $B_m$ as a theta average, the rational Weyl element, a local Weyl calculation, the full reflected expansion, and the quadratic large sieve for the exterior symbols (Lemma 6.4, verified from Goldmakher--Louvel's Theorem 1.1 and Heath-Brown's real-character mean value). The sextic exponents $1$ and $-4$ sum to $3\pmod 6$, making the pairing quadratic.
• Section 7, The Poisson probe and its Euler product (pp. 41--51; 05-poisson.tex). Weights $W_0,W_1$, a separated additive large sieve (Lemma 7.1), the probe bound $I(Z^a,Z^b,Z)\ll Z^{933/2000+\epsilon}$ (Section 7.1), Poisson summation in three Mellin variables $(\xi,w,z)$ (Section 7.2), and the Euler factorization of the frequency series $\mathcal{F}_u$ with a correction $\mathcal{H}_u$ holomorphic in two explicit numerical regions and bounded by $q_u^\epsilon$; $\mathcal{H}_1$ is bounded and nowhere zero once $S$ contains all primes of norm below a fixed constant (Section 7.3).
• Section 8, Zero detection and Mellin continuation (pp. 51--60; 06-zero-free.tex). Lemma 8.1, a primitive conductor large sieve over $F$; a zero-density proposition (Proposition 8.2, p. 53) putting $\ll U^{1/2}$ rows in an exceptional set and bounding reciprocals of the remaining $L$-functions in $\operatorname{Re}\xi\ge.998$; the principal row and the other frequency rows (Section 8.2); completion (Section 8.3), with Table 1 (p. 59) listing the seven numerical exponent margins relative to $7/15$ (the one nearest zero, $-1/6000$, from the reflection bound), the bound $f_\eta(Z)\ll Z^{7/15-1/20000}$, the Mellin transform $\mathcal{T}(s)$ holomorphic for $\operatorname{Re}s>1-1/20000$, its identification with $e^{(s-5/6)^2}\mathcal{H}_\eta(s)/L^S(s,\eta)$ by Fourier inversion and the identity theorem, and the conclusion of Theorem 1.2 (the half-plane the manuscript claims contains $\operatorname{Re}s>1-10^{-6}$).
• Section 9, A uniform estimate for complete splitting (pp. 60--63; 07-splitting.tex). Lemma 9.1: for large $q$, $[K_q:\mathbb{Q}]=q(q-1)$, $\log D_q+n_q\ll_a n_q\log(2q)$, and $\zeta_{K_q}$ has no zero in $\operatorname{Re}s>1-\delta_0$, $s\ne1$, by placing $K_q$ under $\widetilde K_q=K_q\mathbb{Q}(\mu_{12q})$, factoring $\zeta_{\widetilde K_q}$ into Hecke $L$-functions of $\mathbb{Q}(\mu_{12q})$ covered by Theorem 1.2, and descending (Neukirch, Chapter VII). Lemma 9.2: a smooth explicit formula $\Theta_{K,f}(x)\ll_f x+x^{1-\delta}(\log D+n)$ for any number field whose zeta function is zero-free in $\operatorname{Re}s>1-\delta$, with the Tate functional equation and the Hasanalizade--Shen--Wong zero count. Proof of Proposition 2.2 (p. 63).
• Section 10, Marked Type II estimates and rough factors (pp. 63--70; 08-type-ii.tex). $W=\exp(L^{0.24})$, prime groups $\mathcal{P}_i=[\exp(L^{a_i}),\exp(2L^{a_i})]$ with $0.1<a_1<\cdots<a_K<0.2$, the marked weight $\mathcal{W}(h)$. Theorem 10.1 (one-sided marked Type II estimate) is stated as imported from the Poisson--Dirichlet release manuscript's Theorem 3.1, without proof here. Lemma 10.2 (coefficient criterion) is, in the manuscript's words, extracted from the proof of that cited theorem: it depends on the structure of the cited argument, not only on its statement. Proposition 10.3 verifies the criterion for the rough-factor coefficient $\mathbf{1}_{P^-(m)>x^\gamma}-D_\gamma(\log m/L)\mathbf{1}_{P^-(m)>W}/(LV(W))$.
• Section 11, Two elementary sieve facts (pp. 70--74; 11-sieve.tex). Lemma 11.1, a Brun--Hooley block sieve with bounded coefficients (Ford--Halberstam product inequalities); Lemma 11.2, Buchstab's rough-number density identity $\int_b^{1/2}D_t(1-t)\,dt/t=D_b(1)-1$ and the bound $D_b(1)\le e^{-\gamma_E}b^{-1}(1+C_d e^{-c_d/b})$ for small $b>0$, with $c_d=1/20$ allowed. Both adapted, with proofs, from the weighted-dilation release manuscript.
• Section 12, Primes with a controlled predecessor in a fixed progression (pp. 74--84; 12-construction.tex). Proves Proposition 2.1: weights $w(d)=\mathbf{1}_{d\equiv u (M)}\Psi(d/x)F(d-1)\mathcal{W}(d-1)$ with $F$ detecting $h/h_{\mathcal{P}}=cQ$; a Type II estimate in the progression (Lemma 12.1, from Lemma 10.2 and Proposition 10.3 after expanding the congruence in Dirichlet characters); the harmonic mass $J_0$ (Lemma 12.2); distribution of the weighted family and $X_0\asymp xJ_0/L$ by Bombieri--Vinogradov (Lemma 12.3); mass conditioned on a divisor (Lemma 12.4); removal of composites by their least prime factor; an upper bound for pairs of nearly equal prime factors (Lemma 12.5, constant independent of $K$); and the parameter choice $\kappa<0.01$, $\epsilon<\kappa$, $b$ small, then $K$ large and $a_i=0.1+0.1i/(K+1)$, closing with prime mass at least $(0.9-0.2-1/4+o(1))\mathfrak{S}_MX_0/L$.
• Appendix A, The cubic theta normalization (pp. 84--90; 10-whittaker-calculation.tex). The Kazhdan--Patterson cover with $c=2$, the central datum, the normalized global Whittaker expansion, the spherical induced vector, the shells of the Jacquet integral and the unit normalization behind Proposition 5.1; cites Kazhdan--Patterson 1984 with its 1985 corrections and Kubota.
• References (pp. 91--92; references.tex): 30 entries, including four release manuscripts (the Poisson--Dirichlet law, the weighted dilation graphs, prime predecessors with an even number of prime factors, and the Quasi-Riemann Hypothesis).

Flags. The manuscript marks nothing as conditional, numerical or computer-assisted; its analytic conclusion rests on the hand-tabulated exponent margins of Table 1, and its sieve conclusion on the imported Theorem 10.1 and on a criterion extracted from that theorem's proof in another unreviewed release manuscript. The release folder holds no verification/ directory for this manuscript.

Bears on

• Problem 985: background. The question asks whether every prime $p$ has a prime primitive root $q<p$. Theorem 1.1 with $a=2$ claims at least $c_2x/(\log x)^2$ primes $p\in(x,2x)$ with the prime $2$ as a primitive root, so infinitely many $p$ would have a prime primitive root below $p$; it says nothing about the "every $p$" quantifier, and Heath-Brown's 1986 theorem, cited on the page, already gives infinitely many such $p$ with one of $2,3,5$. The claim is unverified here, and the page's status rests on its own acceptance evidence.
• Problem 429: comparison with an input the page records as cited and not held. Weisenberg's Theorem 1, the page's status-defining source, fixes a positive integer that is a primitive root modulo infinitely many primes and cites Gupta--Murty and Heath-Brown for its existence. Theorem 1.1 would, if accepted, name every positive admissible base (every integer $a\ge2$ that is not a square, for instance $a=2$) as such an integer, with a count; the disproof does not depend on it, since the cited classical results already supply existence and the paper's second construction avoids the input. The claim is unverified here, and the page's status rests on the refereed source it names.