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Math-2774 · mathematics · Number theory

Primitive-root infinitude for every admissible base

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every integer a that is neither -1 nor a perfect square, are there infinitely many primes p not dividing a for which a generates the multiplicative group modulo p?

Short English statement

For every integer a that is neither -1 nor a perfect square, are there infinitely many primes p not dividing a for which a generates the multiplicative group modulo p?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2774-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2774

For each integer $a$ that is neither $-1$ nor a square, the source gives at least $c_a x/(\log x)^2$ suitable primes in every sufficiently large interval $(x,2x)$. This proves the infinitude requested here and adds a quantitative lower bound. It does not assert the positive prime-relative density required by the separate Artin density conjecture.

Primitive-root infinitude for every admissible base

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 029; 2026-10-04), the asserted result is that each integer other than $-1$ or a square is primitive modulo at least $c_a x/(\log x)^2$ primes between $x$ and $2x$ for large $x$.

Mathematical statement of the claimed result

Manuscript 1.
An integer $a$ is admissible when $a\ne-1$ and it is not an integer square. For each such $a$, the claim supplies $c_a>0$, $x_a\ge2$ such that every real $x\ge x_a$ satisfies \[ |\{p\text{ prime}:x<p<2x,\ p\nmid a,\ \operatorname{ord}_p(a)=p-1\}| \ge c_a x/(\log x)^2. \] Here $\operatorname{ord}_p(a)$ is multiplicative order. The estimate yields infinitely many primitive-root primes for each admissible base, without asserting Artin's predicted positive-density asymptotic.

Outline of the argument

For manuscript 1 (family 029; 2026-10-04), cubic theta coefficients under Kummer twists give a Hecke zero-free comparison. Controlled prime predecessors in fixed progressions are then sieved against complete-splitting obstructions, retaining enough primes on which the base has full multiplicative order.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 029; 2026-10-04).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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