On gaps between quadratic non-residues in the Euclidean and Hamming metrics

Rainer Dietmann, Christian Elsholtz, Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


The authors have recently introduced and studied a modification of the classical number theoretic question about the largest gap between consecutive quadratic non-residues and primitive roots modulo a prime p, where the distances are measured in the Hamming metric on binary representations of integers. Here we continue to study the distribution of such gaps. In particular we prove the upper bound ℓp≤(0.117198...+o(1))logp/log2 for the smallest Hamming weight ℓp among prime quadratic non-residues modulo a sufficiently large prime p. The Burgess bound on the least quadratic non-residue only gives ℓp≤(0.15163...+o(1))logp/log2.

Original languageEnglish
Pages (from-to)930-938
Number of pages9
JournalIndagationes Mathematicae
Issue number4
Publication statusPublished - 15 Nov 2013


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