Result 029, Number theory

Primitive roots for every admissible integer base

Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least cax/(log⁡x)2c_a x/(\log x)^2 primes in every sufficiently large interval (x,2x)(x,2x) have primitive root a, with ca>0c_a\gt 0.

Proof

The bigger picture

Why it matters

Can powers of one fixed integer cycle through every nonzero remainder modulo infinitely many primes? The manuscript claims they can for every eligible integer, linking a simple multiplication rule to the distribution of primes.

What changes?

The unreviewed manuscript reports this for every integer a that is neither -1 nor a square. Calling a a primitive root modulo a prime means its powers produce every nonzero remainder modulo that prime. For every sufficiently large x, it claims at least c_a times x divided by the square of log x such primes between x and 2x. The positive constant c_a and the threshold for x may depend on a. This establishes the conjecture's infinitude assertion, not an exact density.

What does that help mathematicians do?

The claimed bound would show more than an endless supply of suitable primes: every sufficiently large interval from x to 2x would contain a quantitatively guaranteed supply for each fixed eligible base. A companion manuscript extends that question to every fixed finite set of distinct positive prime bases, with the same form of lower bound for primes where all are primitive roots. That simultaneous claim is conditional on four analytic and sieve inputs.

Are there practical applications?

The immediate value is foundational: the result concerns when repeated multiplication achieves the longest possible cycle among nonzero remainders modulo a prime. Its quantitative guarantee would give researchers a constraint on how often these maximal cycles occur for a fixed base. The supplied abstracts do not describe a practical algorithm for finding the primes.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Primitive roots for every admissible integer base

October 4, 2026 92 pages

We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least cax/(log⁡x)2c_a x/(\log x)^2 primes in (x,2x)(x,2x) with primitive root a, for some ca>0c_a\gt 0 and every sufficiently large x.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Simultaneous primitive roots: a conditional lower bound for prime bases

October 4, 2026 21 pages

For every fixed finite set of distinct positive primes, we prove that at least cx/(log⁡x)2cx/(\log x)^2 primes in (x,2x)(x,2x) have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.