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

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 pp, 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/log2ℓp≤(0.117198…+o(1))logp/log2 for the smallest Hamming weight ℓpℓp among prime quadratic non-residues modulo a sufficiently large prime pp. The Burgess bound on the least quadratic non-residue only gives ℓp≤(0.15163…+o(1))logp/log2ℓp≤(0.15163…+o(1))logp/log2.