Quadratic non-residues in short intervals

Sergei V. Konyagin, Igor E. Shparlinski

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes p in a dyadic interval [Q, 2Q] for which a given interval [u + 1, u + ψ(Q)] does not contain a quadratic non-residue modulo p. The bound is non-trivial for any function ψ(Q)→ ∞ as Q→ ∞. This is an analogue of the well-known estimates on the smallest quadratic nonresidue modulo p on average over primes p, which corresponds to the choice u = 0.

Original languageEnglish
Pages (from-to)4261-4269
Number of pages9
JournalProceedings of the American Mathematical Society
Issue number10
Publication statusPublished - 1 Oct 2015
Externally publishedYes


  • Character sums
  • Quadratic non-residues


Dive into the research topics of 'Quadratic non-residues in short intervals'. Together they form a unique fingerprint.

Cite this