Vinogradov's least quadratic nonresidue conjecture

About 5 years old · traced to 2021

Let $p$ be a prime, and let $n(p)$ denote the least natural number that is not a quadratic residue modulo $p$ . For any fixed $\epsilon>0$ , the least quadratic nonresidue satisfies

n(p)\ll p^{\epsilon}.

This conjecture asserts that the least quadratic nonresidue is smaller than every fixed positive power of $p$ , up to an implied constant depending on $\epsilon$ . Vinogradov established the bound $n(p)\ll p^{1/\sqrt{2e}}\log^2 p$ , but the conjecture remains open unconditionally.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Vinogradov's least quadratic nonresidue conjecture
For each prime $p$ , let $n(p)$ denote the least natural number that is not a quadratic residue modulo $p$ . Vinogradov's conjecture. For any fixed $\varepsilon>0$ , we have
$n(p) \ll p^\varepsilon.$
This conjecture follows from the generalised Riemann hypothesis and is known to hold for almost all primes $p$ , but remains open unconditionally in general; the best bound available up to logarithmic factors is due to Burgess.
source: Terence Tao, “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture”, arXiv:1410.7073 (2015).

References

Primary source

Shivarajkumar, “Vinogradov's Conjecture and Beyond”, arXiv:2111.05219 (2022).

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