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SHORTEST DISTANCE IN MODULAR HYPERBOLA AND LEAST QUADRATIC NON-RESIDUE

Part of: Elementary number theory Exponential sums and character sums

Published online by Cambridge University Press:  10 May 2016

Tsz Ho Chan
Tsz Ho Chan*
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A. email thchan6174@gmail.com
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Abstract

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c\;(\text{mod}\;p)$ . There are two such points in a square of side length $p^{1/4+\unicode[STIX]{x1D716}}$ . Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\unicode[STIX]{x1D716}}$ or the least quadratic non-residue is less than $p^{1/(6\sqrt{e})+\unicode[STIX]{x1D716}}$ .

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems 11A15: Power residues, reciprocity 11L40: Estimates on character sums
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 860 - 865
Copyright
Copyright © University College London 2016 

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