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An Unconditional Artin-Matthews Theorem For Simultaneous Primitive Roots

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Vlado Vojdanovski ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Let S be a finite set of integers different from 0 and ±1. We prove that the primes for which every member of S is a primitive root have positive lower relative density precisely when neither of Matthews’ two obstructions occurs. The proof uses the marked estimates for prime predecessors and uniform splitting estimates of OpenAI. Its principal step is a refinement of the large-prime-factor law for prime predecessors by a fixed cubic Frobenius condition. We obtain this refinement from a shifted bilinear estimate for cubic theta series, a divisor-weighted positive sieve estimate, and a prime-extraction argument that retains the natural Hecke coefficients. The resulting lower bound holds in every sufficiently large dyadic interval.

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This work was produced as a guided exploration by OpenAI Astra agents.

Formalizations

References

  1. E. Assing, V. Blomer, and J. Li, Uniform Titchmarsh divisor problems , Adv. Math. 393 (2021), Article 108076. doi:10.1016/j.aim.2021.108076 (https://doi.org/10.1016/j.aim.2021.108076). The section and theorem numbers cited here refer to arXiv:2005.13915v1 (https://arxiv.org/abs/2005.13915v1).arXivDOI
  2. M. Anwar and F. Pappalardi, On simultaneous primitive roots , Acta Arith. 180 (2017), no. 1, 35--43. doi:10.4064/aa8566-3-2017 (https://doi.org/10.4064/aa8566-3-2017).DOI
  3. M. Bhargava, Every nonsquare integer not equal to −1-1 is a primitive root modulo a positive lower density of primes , Hexagon preprint 2610.00228v1, 14 October 2026. Public manuscript (https://hexagonmath.org/2610.00228v1).hexagon
  4. K. R. Matthews, A generalisation of Artin's conjecture for primitive roots , Acta Arith. 29 (1976), no. 2, 113--146. doi:10.4064/aa-29-2-113-146 (https://doi.org/10.4064/aa-29-2-113-146).DOI
  5. T. Miyake, Modular Forms , Springer Monographs in Mathematics, Springer, Berlin, 2006. doi:10.1007/3-540-29593-3 (https://doi.org/10.1007/3-540-29593-3).DOI
  6. OpenAI, Primitive roots for every admissible integer base , preprint, 4 October 2026. Public manuscript (https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf).link
  7. OpenAI, The Poisson--Dirichlet law for prime predecessors , preprint, 24 September 2026. Public manuscript (https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf).link
  8. OpenAI, Weighted dilation graphs, smooth shifted primes and totient fibers , preprint, 24 September 2026. Public manuscript (https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf).link
  9. OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜs>7/8\Re s>7/8 , preprint, 30 September 2026. Public manuscript (https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf).link

Version history

  1. v1 · 25 pagesSubmitted by Vlado VojdanovskiInitial depositCurrentNov 11, 2026