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Forum Mathematicum

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Volume 29, Issue 2


On the smallest simultaneous power nonresidue modulo a prime

Kevin Ford
  • Department of Mathematics, 1409 West Green Street, University of Illinois at Urbana–Champaign, Urbana, IL 61801, USA
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/ Moubariz Z. Garaev
  • Corresponding author
  • Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, Mexico
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/ Sergei V. Konyagin
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/forum-2015-0250


Let p be a prime and let p1,,pr be distinct prime divisors of p-1. We prove that the smallest positive integer n which is a simultaneous p1,,pr-power nonresidue modulo p satisfies


for some positive cr satisfying cr=e-(1+o(1))r as r.

Keywords: Simultaneous power nonresidues; primitive roots; sieve methods; well-spaced divisors

MSC 2010: 11A15; 11A07; 11N29


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About the article

Received: 2015-12-07

Revised: 2016-05-16

Published Online: 2016-09-14

Published in Print: 2017-03-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1201442

Award identifier / Grant number: DMS-1501982

Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 14-01-00332

The first author is supported in part by the National Science Foundation grants DMS-1201442 and DMS-1501982. The third author is supported by grant RFBR 14-01-00332.

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 347–355, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0250.

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