Vinogradov's least quadratic nonresidue conjecture

For an odd prime $p$ , let $n(p)$ be the least positive integer that is a quadratic nonresidue modulo $p$ . Unconditionally, the best bounds (Burgess with Vinogradov's trick) give $n(p)\ll_\varepsilon p^{1/(4\sqrt e)+\varepsilon}$ , while the generalized Riemann hypothesis gives $n(p)\ll(\log p)^2$ . Vinogradov conjectured that $n(p)\ll_\delta p^\delta$ for every $\delta>0$ . Is it true that for every $\delta>0$ the least quadratic nonresidue satisfies $n(p)\ll_\delta p^\delta$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Analytic number theory
Posed by: I. M. Vinogradov, as stated in Tao, The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture (2015), Conjecture 1.1
Year posed: —
Years open: —
Solved: 2026-09-30
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 45 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims absolute constants $A,C>0$ with $n(p)\le C(\log p)^A$ for every odd prime $p$ , hence $n(p)\ll_\delta p^\delta$ for every $\delta>0$ . This is a polylogarithmic bound, much stronger than the conjecture asks, though weaker than the $(\log p)^2$ that GRH gives. The same corollary gives a deterministic polynomial-time algorithm for square roots modulo a prime. The proof is conditional on nothing beyond the family's own zero-free half-plane $\Re s>7/8$ and the published theorem of Bhargava, Ivanyos, Mittal and Saxena; it is not an independent argument.

What the AI did

The release README says every manuscript in the collection was produced by an unreleased internal OpenAI model, and that the vast majority were obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result. It names the zero-free region for the Riemann zeta function as an exception to that fixed procedure, without saying what was done instead, and says the write-up of the companion Re(s) > 11/12 proof was human edited for readability. The manuscripts are credited to OpenAI with no human author named. The nonresidue bound is a corollary in the 7/8 manuscript, derived from the zero-free half-plane through a published theorem of Bhargava, Ivanyos, Mittal and Saxena.

Verification

No independent mathematician has checked this yet. Corollary 1 of the quasi-Riemann manuscript was read against the posed conjecture: it claims $n(p)\le C(\log p)^A$ with absolute constants (for example $A=32$ ), which implies Vinogradov's conjecture. The deduction is a few lines: the 7/8 half-plane and the functional equation put all nontrivial zeros of primitive Dirichlet L-functions in a strip that supplies the weak-GRH hypothesis of Bhargava, Ivanyos, Mittal and Saxena (ISSAC 2017, Conjecture 6.3 and Theorem 6.7), whose conclusion gives the bound. The release's Lean main results for this manuscript state only the zero-free half-plane; neither the nonresidue bound nor the cited BIMS theorem is formalized, so this entry is not marked Lean-checked. It stands or falls with the quasi-Riemann claim.

Vinogradov's least quadratic nonresidue conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion