Problem 985

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Statement. Is it true that, for every prime $p$, there is a prime $q<p$ which is a primitive root modulo $p$?

Statement (corrected). Is it true that, for every prime $p>2$, there is a prime $q<p$ which is a primitive root modulo $p$?

Notes. The site's wording quantifies over every prime $p$ and fails at the smallest one: no prime is smaller than $2$, so $p=2$ has no prime primitive root $q<p$. A comment of 17 August 2025 by Woett in the site's discussion thread makes this remark, that $p>2$ is required, and the formal-conjectures statement assumes $p\ne2$. The change inserts "$>2$" after "every prime $p$"; nothing else changes. The defect is already in the poser's text: Erdős asks the question for every prime with no restriction, in [Er61e], p. 11, and in [Er65b], printed p. 233: "As far as I know it is not even known whether to every $p$ there is a prime $q<p$ which is a primitive root of $p$." The problem's standing judges the corrected Statement.

Formulation. The site's wording is Erdős's question as he asks it in [Er65b] (printed p. 233), the site's source, and in his 1961 note [Er61e] (p. 11). The corrected Statement asks that the least prime primitive root of every prime $p>2$ be smaller than $p$; Artin's conjecture, which the site's commentary cites, concerns a fixed base instead, and asks for infinitely many primes $p$ to which that base is a primitive root.

Status. Open. The site's label is OPEN, which describes the corrected Statement. No claim page is recorded.

Source. erdosproblems.com/985, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #985, https://www.erdosproblems.com/985.

References.

• [Er61e] Erdős, P., Számelméleti megjegyzések I (Remarks on number theory I). Mat. Lapok 12 (1961), 10--17; p. 11. Library home: erdos_1961_szamelmeleti_megjegyzesek.
• [Er65b] Erdős, P., Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III, Wiley (1965), 196-244; printed p. 233. Library home: erdos_1965_recent_advances_current_problems_number_theory.
• [He86b] Heath-Brown, D. R., Artin's conjecture for primitive roots. Quart. J. Math. Oxford Ser. (2) (1986), 27-38.
• [Ho67b] Hooley, Christopher, On Artin's conjecture. J. Reine Angew. Math. (1967), 209-220.

Formalization. Statement in formal-conjectures.

Current assessment

Scope. The page carries the site's label (OPEN, accessed 2026-09-04), which describes the corrected Statement, and its references on Artin's conjecture; no status search is recorded. The notes below record the literature on the least prime primitive root and the release's manuscripts that bear on the question.

The least prime primitive root. Under the generalized Riemann hypothesis, Shoup (Searching for primitive roots in finite fields, Math. Comp. 58 (1992), 369-380) bounds the least primitive root of a prime $p$ by $\ll(\omega(p-1)\log\omega(p-1))^4(\log p)^2$, and Martin (The least prime primitive root and the shifted sieve, Acta Arith. 80 (1997), 277-288, Corollary 3.1) rederives this bound, which he attributes to Shoup for prime moduli, for the least prime primitive root, which is therefore below $p$ for every sufficiently large $p$. Unconditionally, Nongkynrih (On prime primitive roots, Acta Arith. 72 (1995), 45-53) bounds the least prime primitive root by $(\log p)^{O(\log_3 p/\log_4 p)}$ for almost all $p$, and Martin's Theorem 1 improves this to a fixed power of $\log p$ for all prime powers up to $Y$ outside a set of $O(Y^\varepsilon)$ of them. Neither result has a claim page: the conditional bound carries an inexplicit constant, so it covers only primes beyond an uncomputed threshold, and the almost-all bounds name no prime, so neither settles an instance of the question. Computations of the least prime primitive root up to $10^{14}$ reported on the site's thread cite no published source, so they are not recorded here.

The release on Artin's conjecture. The OpenAI mathematics release's manuscript Primitive roots for every admissible integer base (4 October 2026; folder preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026 of github.com/openai/math, pinned by that link) states that every integer $a$ other than $-1$ and the squares is a primitive root modulo at least $c_a x/(\log x)^2$ primes in $(x,2x)$ for all large $x$, the infinitude part of Artin's conjecture for every admissible base; of the references above, [Ho67b] proves that conjecture under the generalized Riemann hypothesis and [He86b] shows that it can fail for at most a few prime or squarefree bases. It is a fixed-base statement and background to this problem: the question here quantifies over every prime $p>2$ and asks for some prime primitive root below $p$, which the manuscript does not address. The release lists no Lean for it; its card is openai_2026_primitive_roots_admissible_integer_base, whose note for this problem records it as background, nothing in it is verified in this corpus, and it has no claim page.

The release's zero-free regions. The release's manuscripts The Quasi-Riemann Hypothesis (30 September and 5 October 2026) and Uniform exclusion of Landau-Siegel zeros (1 October 2026), with the cards openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8, openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12 and openai_2026_uniform_exclusion_landau_siegel_zeros, claim zero-free half-planes for Dirichlet $L$-functions and an exclusion of Landau-Siegel zeros. The two Quasi-Riemann cards link this problem because such a half-plane is the kind of input that the conditional bound above takes from the generalized Riemann hypothesis, and the Landau-Siegel card records that its theorem does not apply here, since a prime primitive root below $p$ calls for bounds on character sums over primes that the theorem does not give; the manuscripts state no result on this problem, no deduction from them is recorded here, and they have no claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1961_szamelmeleti_megjegyzesek
• erdos_1961_szamelmeleti_megjegyzesek / problem_p11
• openai_2026_primitive_roots_admissible_integer_base
• openai_2026_primitive_roots_admissible_integer_base / theorem_1_1
• openai_2026_primitive_roots_admissible_integer_base / theorem_1_2
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12 / corollary_1_2
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12 / theorem_1_1
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8
• openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8 / theorem_1_1
• openai_2026_uniform_exclusion_landau_siegel_zeros