∑math20

Number theory · Result 029

Primitive roots for every admissible integer base

2 papers

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 \(c_a x/(\log x)^2\) primes in every sufficiently large interval \((x,2x)\) have primitive root a, with \(c_a\gt 0\).

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Primitive roots for every admissible integer base

Primitive roots for every admissible integer base

OpenAI · October 4, 2026

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 \(c_a x/(\log x)^2\) primes in \((x,2x)\) with primitive root a, for some \(c_a\gt 0\) and every sufficiently large x.

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Cite
@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}
}
First page of Simultaneous primitive roots: a conditional lower bound for prime bases

Simultaneous primitive roots: a conditional lower bound for prime bases

OpenAI · October 4, 2026

For every fixed finite set of distinct positive primes, we prove that at least \(cx/(\log x)^2\) primes in \((x,2x)\) 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.

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Cite
@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.