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Math-1504 · mathematics · Graph theory

O19 — Finding a primitive root modulo a prime

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $p$ be a prime. A primitive root modulo $p$ is an integer $g$ with $1\le g<p$ such that \[ \{g^j\bmod p:0\le j<p-1\}=\{1,\ldots,p-1\}. \] Equivalently, $g$ has multiplicative order $p-1$ in $\mathbb F_p^\times$. Such an element exists because this finite multiplicative group is cyclic; for $p=2$ the value $g=1$ qualifies.

Does there exist a deterministic classical algorithm which, on every binary prime input $p$, outputs a primitive root using a number of bit operations bounded by a fixed polynomial in $\ell(p)=\lfloor\log_2p\rfloor+1$? The output may be any primitive root, with no minimality requirement.

Only $p$ is provided. In particular, a complete factorization of $p-1$, useful for checking candidate orders, is not included in the input. All preliminary computations needed for the construction are charged to the same running-time bound. The primality of $p$ may be treated as a promise.

The task is stronger than producing a polynomially sized list known merely to contain a primitive root, unless the algorithm can identify a successful member within the required time. A proof that many residues generate the group also does not itself give deterministic selection. The aim is an unconditional uniform search procedure for individual arbitrary primes, in the classical bit model.

Short English statement

Can a primitive root modulo every prime be found deterministically in time polynomial in the prime’s binary length?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1504-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1504

Repeated quadratic factorization yields an element with maximal two-primary order, which generates when the multiplicative-group order is a power of two. The requested generator for primes with additional odd-order factors is not obtained by that argument.

O19 — Finding a primitive root modulo a prime

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 142; 2026-10-04), the asserted result is that dense polynomials over binary-specified prime fields have uniform deterministic complete factorization in bit time polynomial in degree and $\log p$, using the cited Hecke zero-free input.

Mathematical statement of the claimed result

Manuscript 1.
For any binary-encoded prime $p$ and nonzero densely encoded $f\in\mathbb F_p[x]$ of degree $n$, a uniform deterministic algorithm is claimed to output \[ f=c\prod_i g_i^{e_i} \] with $c$ the leading coefficient, distinct monic irreducible $g_i$ and positive integer multiplicities, in $O(((n+1)\lceil\log_2p\rceil)^{10^{12}})$ bit operations. Constants have an empty factor list. The computation is unconditional and deterministic; it receives neither an integer-factorization service nor a primitive-root oracle.

The separate algebraic reduction sets $L=\lceil\log_2p\rceil$, $B=20+(n+1)(L+1)$. If $p>B^{200000}$ and $n\ge2$, supply for every prime $q\le n$ a prime $\ell_q\notin\{2,3,p,q\}$ with \[ \ell_q\equiv1\pmod{12q},\qquad p^{(\ell_q-1)/q}\not\equiv1\pmod{\ell_q}. \] That reduction has bit cost $O((B+E)^C)$ for absolute $C$, where $E=\max(2,\max_q\ell_q)$, and $E=2$ when no such primes are required. This bound depends on the numerical size of the auxiliary primes; the full theorem additionally constructs sufficiently small primes.

Outline of the argument

For manuscript 1 (family 142; 2026-10-04), primary generators and norm equations reduce splitting to an odd-degree separator. Lifted ramification classes are filtered to keep extension degrees polynomial, and divisors at infinity distinguish roots; the analytic input bounds the auxiliary primes required by the finite algebraic algorithm.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 142; 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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