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arxiv: 2411.16000 · v41 · submitted 2024-11-24 · 🧮 math.NT

Global Primitive Roots of Unity

Wayne Lewis This is my paper

Pith reviewed 2026-05-23 08:21 UTC · model grok-4.3

classification 🧮 math.NT
keywords primitive rootsArtin conjectureultrafiltersultraproductsnatural densityBézout domainsChebotarev theoremadelic tori
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The pith

A nonprincipal ultrafilter on the primes produces infinitely many p where (p-1)/6 is prime and m is a primitive root modulo p for non-square m not equal to -1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a Bézout subdomain from the ultraproduct of the p-adic integers over a nonprincipal ultrafilter on the primes. This yields a valued field whose relative algebraic closure satisfies specific root and torsion conditions derived from Chebotarev's theorem. Using this, it proves there are infinitely many primes p satisfying the two conditions on (p-1)/6 being prime and m being primitive root. It also derives the natural density of the set of such primes as an entangled Artin product without using GRH. A reader would care because these results give unconditional resolutions to conjectures on primitive roots that have been open for decades.

Core claim

The central discovery is that the Bézout subdomain of the valuation domain with respect to a suitable nonprincipal ultrafilter, together with the relative algebraic closure L of the prime field in the ultraproduct, where L contains sqrt(-p~) for all p, no cube roots of q~, and tor(L^x) = <zeta_6>, allows positive resolution of the infinitude of primes p with (p-1)/6 prime and m primitive root mod p, and the density of T_m equals c_m times the product over q of (1 - 1/(q(q-1))).

What carries the argument

The nonprincipal ultrafilter on the primes and the resulting relative algebraic closure L with controlled roots and torsion group generated by a sixth root of unity.

If this is right

  • There are infinitely many primes p with (p-1)/6 prime and m a primitive root mod p.
  • The set T_m has natural density equal to the corrected Artin Euler product without assuming GRH.
  • Ultraproduct methods certify countably infinite conforming sets for number theory problems requiring infinitude.
  • The quantitative Artin primitive root conjecture holds via normalized ergodic Haar measure on the universal adelic torus.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the ultrafilter construction generalizes, similar unconditional proofs may apply to other variants of Artin's conjecture.
  • The approach indicates that nonstandard techniques can replace analytic assumptions like GRH in computing densities of special primes.
  • Links to the adelic torus suggest potential extensions to ergodic questions in broader adelic settings.

Load-bearing premise

There exists a nonprincipal ultrafilter on the set of primes such that the relative algebraic closure of the prime field in the ultraproduct contains the square root of the negative of every prime, contains no cube root of any prime, and has multiplicative torsion group generated by a primitive sixth root of unity.

What would settle it

A proof that no such ultrafilter exists, or an explicit demonstration that only finitely many primes p satisfy both (p-1)/6 prime and m primitive root mod p for some fixed non-square m not -1.

Figures

Figures reproduced from arXiv: 2411.16000 by Wayne Lewis.

Figure 1
Figure 1. Figure 1: Be and Ze {p ∈ P: ((wp + pZp)p∈P) = ((zp + pZp)p∈P)} ∈ v ⇔ {p ∈ P: wp − zp = pyp for some yp ∈ Zp} ∈ v ⇔ ΘZb((wp − zp)p∈P) ∈ s˜Ze ⇔ ΘZb((wp)p∈P) + ˜sZe = ΘZb((zp)p∈P) + ˜sZe. The bottom right triangle commutes by definition of β. The maps qZb1 and qZb2 are quotient maps, so γ is a well-defined isomorphism and the bottom left triangle commutes because (wp)p∈P + sZb = (zp)p∈P + sZb ⇔ (wp + pZp)p∈P = (zp + pZ… view at source ↗
read the original abstract

An ideal setting to exhibit infinite sets of primes $p$ relative to which an integer is a primitive root $\pmod p$ is provided by the B\'ezout subdomain $\widetilde{\mathbb{B}}:=\mathbb{Z}^{\mathbb{P}}/\mathfrak{U}$ of the valuation domain $\widetilde{\mathbb{Z}}=\prod_{\mathfrak{U}} \mathbb{Z}_p$ with respect to a nonprincipal ultrafilter $\mathfrak{U}$ on $\mathbb{P}$, extant via Chebotarev's theorem and the ultrafilter theorem and such that the relative algebraic closure $\mathbb{L}:=\mathrm{Abs}(\widetilde{\mathbb{Q}})$ of the prime field of the valued field $\widetilde{\mathbb{Q}}=\prod_{\mathfrak{U}} \mathbb{Q}_p$ contains $\sqrt{-\tilde p}$ for $p\in\mathbb{P}$, contains no $\sqrt[3]{\tilde q}$ for $q\in\mathbb{P}$, and has $\mathrm{tor}(\mathbb{L}^\times)=\langle \zeta_6\rangle$. Results include positive resolutions of the conjectured infinitude of primes $p$ for which (i) $\frac{p-1}{6}$ is prime and (ii) a non-perfect-square $-1\neq m\in\mathbb{Z}$ is a primitive root $\pmod p$, establishing as manifest the efficacy of ultraproduct treatments in resolving number theory problems requiring certification of countably infinite conforming sets. Furthermore, we extend these results to the quantitative APRC via normalised ergodic Haar measure on the (monothetic) universal adelic torus $\mathrm{Hom}(\mathbb{Q}^{(\mathfrak{c})},\frac{\mathbb{R}}{\mathbb{Z}})$, leveraging B\'ezout rigidity of $\widetilde{\mathbb{B}}$ and the qualitative APRC witness set $T_m = \{ q\in\mathbb{P} \colon m\text{ is a primitive root}\!\pmod{q}\}$ to present a GRH-free computation of the natural density of $T_m$ as the corrected/entangled Artin Euler product $c_m\prod_{q\in\mathbb{P}}(1-\frac{1}{q(q-1)})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript claims that, for a nonprincipal ultrafilter U on the primes chosen so that the relative algebraic closure L of the prime field in the ultraproduct valued field satisfies tor(L^×)=⟨ζ₆⟩, contains √(-~p) for every prime p, and contains no cube roots of any ~q, the Bézout subdomain ~B transfers the primitive-root condition to show that the set T_m of primes for which a fixed nonsquare m≠-1 is a primitive root is infinite (in fact contains infinitely many p with (p-1)/6 also prime); it further claims a GRH-free computation of the natural density of T_m as the entangled Artin product c_m ∏_{q}(1-1/(q(q-1))) via normalized Haar measure on the universal adelic torus Hom(Q^(c),R/Z).

Significance. If the ultrafilter existence and transfer arguments were valid, the work would supply unconditional infinitude results for a family of Artin-type primitive-root problems and a new ergodic proof of the Artin density formula. The manuscript supplies no machine-checked proofs, reproducible code, or parameter-free derivations that could be credited as independent strengths.

major comments (3)
  1. [Abstract / definition of L] Abstract and the paragraph following the definition of L: the existence of a nonprincipal ultrafilter U such that L simultaneously satisfies tor(L^×)=⟨ζ₆⟩, √(-~p)∈L for every standard prime p, and the absence of all cube roots ~q is asserted to follow from Chebotarev plus the ultrafilter theorem, but no explicit Chebotarev conjugacy classes or splitting conditions are exhibited that enforce all three algebraic constraints at once while keeping the primitive-root witness set nonempty in the ultraproduct. Without this, membership of T_m in U (hence infinitude) does not follow.
  2. [Quantitative APRC paragraph] The density computation via the universal adelic torus: the normalized ergodic Haar measure of the witness set T_m is stated to equal the classical corrected Artin product c_m ∏ (1-1/(q(q-1))), but the argument reduces to identifying the measure with the already-known conjectural expression rather than deriving the Euler product from the ultraproduct construction or from Bézout rigidity of ~B.
  3. [Transfer via ~B] The transfer step from L to T_m ∈ U: the manuscript invokes Bézout rigidity of ~B to conclude that the qualitative APRC witness set lies in U, yet supplies no verification that the asserted roots and torsion in L actually produce a nonempty set of primes satisfying both the primitive-root condition and (p-1)/6 prime inside the ultraproduct.
minor comments (1)
  1. Notation: the symbol ~p for the ultraproduct element is introduced without an explicit definition of the equivalence relation induced by U.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the referee's report. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / definition of L] Abstract and the paragraph following the definition of L: the existence of a nonprincipal ultrafilter U such that L simultaneously satisfies tor(L^×)=⟨ζ₆⟩, √(-~p)∈L for every standard prime p, and the absence of all cube roots ~q is asserted to follow from Chebotarev plus the ultrafilter theorem, but no explicit Chebotarev conjugacy classes or splitting conditions are exhibited that enforce all three algebraic constraints at once while keeping the primitive-root witness set nonempty in the ultraproduct. Without this, membership of T_m in U (hence infinitude) does not follow.

    Authors: The ultrafilter theorem applied to the filter base consisting of the Chebotarev-positive-density sets for each individual condition (quadratic splitting for each √(-p), non-splitting for each cubic root, and cyclotomic splitting for the precise torsion group) yields a nonprincipal U whose relative algebraic closure L satisfies all three constraints simultaneously. These conditions are compatible in the compositum of the relevant Galois extensions, so the intersection remains positive density; the primitive-root witness set T_m remains nonempty in the ultraproduct precisely because m is fixed nonsquare and the torsion is restricted to ⟨ζ₆⟩, which does not force the order of m to drop. Explicit listing of conjugacy classes is not required for the existence argument, which rests on the general form of Chebotarev's theorem. revision: no

  2. Referee: [Quantitative APRC paragraph] The density computation via the universal adelic torus: the normalized ergodic Haar measure of the witness set T_m is stated to equal the classical corrected Artin product c_m ∏ (1-1/(q(q-1))), but the argument reduces to identifying the measure with the already-known conjectural expression rather than deriving the Euler product from the ultraproduct construction or from Bézout rigidity of ~B.

    Authors: The normalized Haar measure on Hom(Q^(c),R/Z) is the unique translation-invariant probability measure. The set of elements whose image generates the full torus (corresponding to m being a primitive root) has measure given by the product of local densities 1-1/(q(q-1)) because the conditions at each prime q are independent under the product topology; the correction factor c_m accounts for the fixed nonsquare m. The ultraproduct supplies the ergodic realization of this measure on the componentwise primes, so the equality is derived from the construction rather than merely identified with the classical formula. revision: no

  3. Referee: [Transfer via ~B] The transfer step from L to T_m ∈ U: the manuscript invokes Bézout rigidity of ~B to conclude that the qualitative APRC witness set lies in U, yet supplies no verification that the asserted roots and torsion in L actually produce a nonempty set of primes satisfying both the primitive-root condition and (p-1)/6 prime inside the ultraproduct.

    Authors: Bézout rigidity of ~B transfers the algebraic relations and multiplicative orders from L back to the primes in U. The torsion condition tor(L^×)=⟨ζ₆⟩ together with the presence of all √(-p) and absence of cube roots ensures that the order of m modulo the ultraprime is exactly p-1, hence m is a primitive root; the same torsion forces (p-1)/6 to behave as a prime in the ultraproduct. The witness set is therefore nonempty inside the ultraproduct by construction of L, placing T_m in U. revision: no

Circularity Check

1 steps flagged

Density 'GRH-free computation' reduces to identifying T_m measure with classical Artin Euler product by construction

specific steps
  1. renaming known result [Abstract (final sentence)]
    "leveraging Bézout rigidity of ˜B and the qualitative APRC witness set T_m = { q∈P : m is a primitive root mod q} to present a GRH-free computation of the natural density of T_m as the corrected/entangled Artin Euler product c_m ∏_{q∈P}(1−1/(q(q−1)))."

    The claimed GRH-free computation is obtained by identifying the ergodic Haar measure on the adelic torus with the already-known conjectural Artin density expression; the ultraproduct construction establishes infinitude of T_m but supplies no new derivation of the explicit Euler product form, so the quantitative result reduces to restating the classical expression.

full rationale

The paper asserts existence of U via Chebotarev+ultrafilter theorem such that L satisfies the three algebraic conditions, then transfers via ~B to conclude T_m ∈ U (hence infinite) and that the natural density equals the Artin product. However, the quantitative claim is obtained by directly equating the Haar measure of the witness set to the pre-existing conjectural Artin expression c_m ∏ (1-1/(q(q-1))), without an independent derivation of that specific product form from the ultraproduct or adelic torus construction. This matches the 'renaming_known_result' pattern: the known conjectural density is relabeled as a new GRH-free computation.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The central claims rest on the asserted existence of a single ultrafilter U whose ultraproduct satisfies three simultaneous algebraic conditions on roots and torsion; these conditions are taken as given via Chebotarev and the ultrafilter theorem, with no independent verification supplied.

axioms (3)
  • ad hoc to paper There exists a nonprincipal ultrafilter U on the set of primes such that the relative algebraic closure L inside the ultraproduct contains √(-p~) for all p, no cube roots, and tor(L^×) = <ζ_6>
    This is the load-bearing modeling assumption invoked to set up the Bézout subdomain and the witness sets T_m.
  • domain assumption Chebotarev's theorem guarantees the required roots and torsion inside the ultraproduct
    Cited as the justification for the algebraic-closure properties of L.
  • standard math The ultrafilter theorem supplies the nonprincipal ultrafilter U
    Standard background fact used to guarantee existence of the ultraproduct.
invented entities (2)
  • Bézout subdomain B~ = Z^P / U inside the valuation domain Z~ no independent evidence
    purpose: To serve as the ambient ring in which infinite conforming sets of primes can be exhibited
    Newly introduced object whose properties are used to certify the infinitude statements.
  • Universal adelic torus Hom(Q^(c), R/Z) equipped with normalised ergodic Haar measure no independent evidence
    purpose: To compute the natural density of the set T_m
    Introduced to extend the qualitative result to a quantitative density formula.

pith-pipeline@v0.9.0 · 5914 in / 2126 out tokens · 36064 ms · 2026-05-23T08:21:24.472523+00:00 · methodology

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