Openai 2026 simultaneous primitive roots conditional lower bound prime bases

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proposition_2_1: The manuscript's prime reservoir, conditional on three inputs it takes from the companion Artin manuscript: many primes p in (x,2x) in a fixed progression with p-1 = 4rQ, Q > x^0.9 prime and r composed of primes in a controlled range. Claims checked only.

theorem_1_1: The manuscript's main claim, conditional on four inputs it states and takes from the companion Artin manuscript: for distinct primes q_1,...,q_k, at least c x/(log x)^2 primes p in (x,2x) have every q_i as a primitive root. Claims checked only.

OpenAI, Simultaneous primitive roots: a conditional lower bound for prime bases, OpenAI Math Release preprint, October 4, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026; the held PDF, simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf in the release, is retained as openai_2026_simultaneous_primitive_roots_conditional_lower_bound_prime_bases.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:Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026,
  author = {{OpenAI}},
  title = {{Simultaneous primitive roots: a conditional lower bound for prime bases}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf}{OAI:Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026}},
  year = {2026}
}

The held PDF has 21 pages with a text layer; its title page is dated October 4, 2026, and its metadata records a creation date of 6 October 2026. The TeX source is build/paper.tex (49 lines), which inputs six section files under build/sections/ and build/references.tex: eight files and 1,640 lines in all, the text read here.

Attestation. The release README states that the collection holds manuscripts "produced by an internal OpenAI model", that it "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" 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 October 4, 2026 and the citation block above, with no sentence on authorship, assistance or verification; the title page and the PDF metadata name OpenAI as the sole author. The abstract and introduction say that the result assumes four inputs whose proofs lie outside the paper. 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 at revision fd4aeeb2e has no entry for it, and its family has no Lean page.

Companions. The release's family for it holds one other manuscript, the October 4, 2026 Primitive roots for every admissible integer base, filed at the Artin card; it claims without hypotheses that every integer other than $-1$ and the squares is a primitive root modulo at least $c_ax/(\log x)^2$ primes in $(x,2x)$ for all large $x$, which for a prime base is the conclusion of Theorem 1.1 here with $k=1$. The release states the relation: this manuscript's abstract and introduction take all four of its Inputs from that manuscript, and Sections 3--5 name the sources, Input 3.1 as its Theorem 1.2, Input 4.1 as the coefficient criterion of its Lemma 10.2 with mark parameter $1/2$ together with the cancellation verified inside the proof of its Proposition 10.3, Input 5.1 as its Lemma 11.1 and Input 5.2 as its Lemma 11.2. The companion's statements of Theorem 1.2, Lemma 10.2, Proposition 10.3, Lemmas 11.1--11.2 and Propositions 2.1--2.2 were read at the same revision for comparison only. Input 3.1 and Inputs 5.1--5.2 agree in substance with Theorem 1.2 and Lemmas 11.1--11.2, with small differences of phrasing (for instance $z>1$ where the companion has $z\ge2$). The companion's Proposition 10.3 states the resulting Type II bound, not the cancellation that Input 4.1(ii) assumes, and the Artin card records that the companion imports its Theorem 10.1 from another release manuscript and extracts Lemma 10.2 from that theorem's proof. On this card's reading, the companion's Propositions 2.1 and 2.2 as stated appear to contain this manuscript's Propositions 2.1 (take the modulus $8M$, $c=4$ and a class $u\equiv5\pmod8$, $u\equiv a\pmod M$) and 2.2 (take the base $a=q$ and the companion's splitting test, Lemma 2.3), so that the companion's claims, if correct, would give Theorem 1.1 here by the argument of Section 2 alone. The manuscript does not make this remark: its introduction presents Sections 4--5 as extending the companion's Section 12 construction to a fixed progression. The comparison was made on the statements only and is unverified. The release lists no other dated version of this manuscript. The companion's arguments were not read for this card.

Read status: claims checked for Theorem 1.1 and Proposition 2.1, together with the statements of Proposition 2.2, the four Inputs and the definitions they use, read clause by clause in the release's TeX source (under build/sections/: 01-introduction.tex label thm:main at lines 32--47; 02-reduction.tex labels prop:reservoir at lines 11--28 and prop:splitting at lines 33--46; 03-splitting.tex label input:hecke at lines 8--15; 04-marked-family.tex lines 10--71 and 88--89 for the marked setup and label input:marked at lines 99--151; 05-prime-reservoir.tex labels input:block at lines 11--63 and input:density at lines 65--78). The proofs (Section 2 for Theorem 1.1, Section 3 for Proposition 2.2, Sections 4--5 for Proposition 2.1) were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The PDF's page numbers are its printed ones; theorems, propositions, lemmas and Inputs share one counter numbered within sections. The four Inputs, which Theorem 1.1 and Propositions 2.1 and 2.2 assume, are summarized after this list and stated in full on the page of Theorem 1.1.

• Abstract and Section 1, Introduction (pp. 1--2; 00-abstract.tex, 01-introduction.tex). Defines $\operatorname{ord}_p(q)$ and primitive roots. Places the one-base problem in Artin's conjecture (recorded by Hasse in 1927, cited to Hasse's diaries) and recalls Hooley's asymptotic under generalized Riemann hypotheses, Matthews's densities for simultaneous primitive roots under the same hypotheses, Anwar and Pappalardi's infinitude criterion under Schinzel's Hypothesis H, and the unconditional results of Gupta and Murty and of Heath-Brown, the latter read as making one member of any three distinct positive primes a primitive root modulo infinitely many primes. Calls the result an implication from four statements of the companion, Inputs 3.1, 4.1, 5.1 and 5.2, reproduced "in the precise forms used here", and states Theorem 1.1 (p. 1): under the four Inputs, for distinct positive primes $q_1,\dots,q_k$ there are $c_{\mathbf q}>0$ and $x_{\mathbf q}>1$ such that for all real $x\ge x_{\mathbf q}$ at least $c_{\mathbf q}x/(\log x)^2$ primes $p\in(x,2x)$ have $p\nmid q_1\cdots q_k$ and $\operatorname{ord}_p(q_i)=p-1$ for every $i$. Outlines the proof: the companion's construction of primes with controlled predecessors, carried into a fixed progression; the marked Type II estimate twisted by a fixed character; Bombieri--Vinogradov in the progression; a direct sieve for balanced semiprimes; a progression making every base a nonresidue; and uniform splitting bounds in the Kummer fields.
• Section 2, Reduction to two estimates (pp. 2--3; 02-reduction.tex). Sets $L=\log x$, $\eta=10^{-6}$ and the order index $\iota_p(q)=(p-1)/\operatorname{ord}_p(q)$. States Proposition 2.1 (a prime reservoir in a fixed progression, assuming Inputs 4.1, 5.1 and 5.2): for odd squarefree $M\ge1$ and a class $a$ with $\gcd(a(a-1),M)=1$ there are $c_{M,a}>0$ and $x_{M,a}>1$ such that for all real $x\ge x_{M,a}$ at least $c_{M,a}x/L^2$ primes $p\in(x,2x)$ have $p\equiv a\pmod M$ and $p=4rQ+1$, with $Q>x^{0.9}$ prime and every prime factor of the positive integer $r$ in $(\exp(L^{0.1}),\exp(L^{0.3}))$. States Proposition 2.2 (a uniform bound for order obstructions, assuming Input 3.1): for each fixed positive prime $q$, all sufficiently large $x$ and uniformly in primes $\ell$ with $\exp(L^{0.1})<\ell<\exp(L^{0.3})$, the primes $p\in(x,2x)$ with $p\nmid q\ell$, $p\equiv1\pmod\ell$ and $q^{(p-1)/\ell}\equiv1\pmod p$ number $\ll_q x/(\ell(\ell-1)L)+x^{1-\eta}$. Proves Theorem 1.1 from the two propositions (pp. 2--3), as summarized on the result page.
• Section 3, A uniform splitting estimate (pp. 3--6; 03-splitting.tex). States Input 3.1, citing the companion's Theorem 1.2, and proves Proposition 2.2: degree $\ell(\ell-1)$ and a discriminant bound for $K=\mathbb Q(\mu_\ell,q^{1/\ell})$ (3.1)--(3.2); the two congruences as complete splitting, by Frobenius; zero-freeness of $\zeta_K$ in $\operatorname{Re}s>1-\eta$ (3.4), by what the text calls the companion's Section 9 tower argument specialized to the base $q$: place $K$ under $K\mathbb Q(\mu_{12\ell})$, factor with abelian Artin formalism (Neukirch), and use that nonprincipal Hecke $L$-functions are entire (Tate); a smoothed explicit formula (3.5) whose error is uniform in the field, with the zero count of Hasanalizade, Shen and Wong (3.6); the upper bound $\Psi_{K'}(x)\ll x+x^{1-\eta}(\log D'+n')$ for the smoothed prime-ideal sum of any number field $K'$ of degree $n'$ and discriminant $D'$ that is zero-free in that half-plane (3.8); and the count (3.9).
• Section 4, A marked family in a fixed progression (pp. 6--12; 04-marked-family.tex). The marked setup and Input 4.1 (see below). Lemma 4.2 (p. 8): the Type II bound of Input 4.1 for the rough-factor coefficient twisted by any character modulo $M$, uniformly in that character. The character expansion of the progression (4.14); the family $w(d)=\mathbf 1_{d\equiv a\,(M)}\Psi(d/x)F(d-1)\mathcal W(d-1)$ (4.15), where $F$ selects $h/h_{\mathcal P}=4Q$ with $Q>x^{0.9}$ prime, and its unique representation $d=4rQ+1$ (4.16). Lemma 4.3 (p. 9): the harmonic mass $J_0=\sum_r\mathcal W(r)/r$ satisfies $1\le J_0\ll_K1$ with a Rankin tail, the prime number theorem in the progression for each small $r$, and $X_0\asymp xJ_0/L$ for the total weight, with constants independent of $K$ and the $a_i$. Lemma 4.4 (p. 11): the sieve remainders are small on average over $r\le x^\epsilon$ and moduli up to $D_0=x^{1/2-\kappa/2}$, by Bombieri--Vinogradov. The sieve products $V_M$ and the constant $\mathfrak S_M>0$ (4.24)--(4.25), and the negligible effect of the group primes dividing $r$ (4.27).
• Section 5, Extracting primes from the marked family (pp. 12--20; 05-prime-reservoir.tex). Inputs 5.1 and 5.2 (see below), with a remark that the density identity removes the least factor. Section 5.1 (p. 14): the initial lower-bound sieve at level $x^b$ (5.5) and a sieve conditioned on a prime divisor $n$ (5.6). Section 5.2 (p. 15): removal of composites by their least prime factor in bins, replacing the rough count by the density through Lemma 4.2 (5.8), and summing the bins with Input 5.2 (5.12)--(5.13). Section 5.3 (p. 17): an upper-bound sieve for balanced semiprimes with a constant independent of $\kappa$, $b$ and $K$ (5.20); the order of the parameter choices; the conclusion (5.21) that primes carry at least $\mathfrak S_MX_0/(2L)$ of the weight, which proves Proposition 2.1 (p. 20).
• References (pp. 20--21; references.tex): 12 entries, one of them the companion release manuscript.

The Inputs. The manuscript assumes four statements of the companion without proof; the page of Theorem 1.1 restates them in full, with the marked setup two of them use. Input 3.1 (p. 3) says that no finite-order Hecke $L$-function of a cyclotomic field containing $\mu_{12}$ vanishes in $\operatorname{Re}s>1-10^{-6}$. Input 4.1 (pp. 7--8) bounds the marked bilinear sum $\sum_{m,n}\alpha_m\beta_nF(mn-1)\mathcal W(mn-1)$ by $XL^{-D_*}$, for $K$ large, whenever the coefficients $\alpha_m$ cancel against Dirichlet characters of modulus up to a power of $L$ and frequencies up to $2XL^B$, and it grants that cancellation to the difference between the indicator of $P^-(m)>x^\gamma$ and its density-weighted model on $W$-rough $m$. Input 5.1 (pp. 12--13) is a block sieve that gives the weight surviving the sifting as its expected main term times $1+O(e^{-H})$, with remainders summed up to level $z^{4H+2}$ and an upper-bound form at $H=2$, all constants depending only on the two bounds assumed for the sieve density $g$. Input 5.2 (p. 13) is the identity $\int_b^{1/2}D_t(1-t)\,dt/t=D_b(1)-1$ for the rough-factor density, which removes the least factor, with the bound $D_b(1)\le e^{-\gamma_E}b^{-1}(1+C_de^{-c_d/b})$ for small fixed $b$.

Flags. The manuscript is conditional by design: Theorem 1.1 and Propositions 2.1 and 2.2 assume the Inputs, which are stated and not proved in it, and by its own account Input 4.1(ii) is a cancellation verified inside a proof in the companion rather than a result stated there. Nothing is marked numerical or computer-assisted; the constants are explicit ($\eta=10^{-6}$ and the exponents $0.1$, $0.24$, $0.3$ and $0.9$) or are fixed in a stated order. 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. The row below is a background lead inferred on this card; it is not stated in the manuscript, it is not verified here, and the page's status rests on its own acceptance evidence. Problem 429, whose recorded disproof uses a positive integer that is a primitive root modulo infinitely many primes, is not listed: that problem is solved, and the classical theorems its page cites already supply that input without hypotheses. Problem 1065, which asks for infinitely many primes $p=2^kq+1$ with $q$ prime, or $p=2^k3^lq+1$, is not listed either. Proposition 2.1 gives primes $p=4rQ+1$ in which every prime factor of $r$ exceeds $\exp((\log x)^{0.1})$, so such a $p$ has one of the asked forms only when $r=1$. The proposition does not promise that case, and its construction gives it weight zero. The proposition therefore controls none of the primes the question asks about; the two share only a theme, the factorization of $p-1$.

• Problem 985: background. The corrected question asks whether every prime $p>2$ has a prime primitive root $q<p$. Under Inputs 3.1, 4.1, 5.1 and 5.2, Theorem 1.1 gives, for any fixed finite set of primes, infinitely many $p$ at which all of them are primitive roots, so each such $p$ has a prime primitive root below it; for a single prime base, $2$ for instance, the companion claims this without hypotheses, and the manuscript's introduction reads Heath-Brown's theorem, cited on the page, as giving it for one of any three distinct primes. The theorem says nothing about the "every $p$" quantifier and bounds the least prime primitive root of no given $p$; no deduction toward the question is known. Unverified.