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Vinogradov's least quadratic nonresidue conjecture

From papers

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

n(p)≪pϵ.n(p)\ll p^{\epsilon}.

This conjecture asserts that the least quadratic nonresidue is smaller than every fixed positive power of pp, up to an implied constant depending on ϵ\epsilon. Vinogradov established the bound n(p)≪p1/2elog⁡2pn(p)\ll p^{1/\sqrt{2e}}\log^2 p, but the conjecture remains open unconditionally.

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Equivalent formulations 1

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

  1. Vinogradov's least quadratic nonresidue conjecture

    For each prime pp, let n(p)n(p) denote the least natural number that is not a quadratic residue modulo pp. Vinogradov's conjecture. For any fixed ε>0\varepsilon>0, we have

    n(p)≪pε.n(p) \ll p^\varepsilon.

    This conjecture follows from the generalised Riemann hypothesis and is known to hold for almost all primes pp, 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).

Sources & referencesView supporting material

Primary source

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

analytic-number-theoryleast-nonresiduesnumber-theoryquadratic-residues

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