Source authenticated

Vinogradov's least quadratic nonresidue conjecture

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 30, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Vinogradov's least quadratic nonresidue conjecture · Vinogradov least nonresidue conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Vinogradov's least quadratic nonresidue conjecture

specialist registry · passed

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.