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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 29, Issue 2

Issues

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

Abstract

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

n<p1/4-cr+o(1)(p)

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

References

  • [1]

    Burgess D. A., The distribution of quadratic residues and non-residues, Mathematika 4 (1957), 106–112. Google Scholar

  • [2]

    Burgess D. A., On character sums and primitive roots, Proc. Lond Math. Soc. (3) 12 (1962), 179–192. Google Scholar

  • [3]

    Burgess D. A., On character sums and L-series. II., Proc. Lond. Math. Soc. (3) 13 (1963), 524–536. Google Scholar

  • [4]

    Erdős P., On the normal number of prime factors of p-1 and some related problems concerning Euler’s φ-function, Quart. J. Math. Oxford Ser. 6 (1935), 205–213. Google Scholar

  • [5]

    Erdős P., On the least primitive root of a prime, Bull. Lond. Math. Soc. 55 (1945), 131–132. Google Scholar

  • [6]

    Erdős P. and Shapiro H. N., On the least primitive root of a prime, Pacific J. Math. 7 (1957), 861–865. Google Scholar

  • [7]

    Halász G., Remarks to my paper: “On the distribution of additive and the mean values of multiplicative arithmetic functions”, Acta Math. Acad. Sci. Hungar. 23 (1972), 425–432. Google Scholar

  • [8]

    Halberstam H. and Richert H.-E., Sieve Methods, Academic Press, New York, 1974. Google Scholar

  • [9]

    Hall R. R. and Tenenbaum G., Divisors, Cambridge Tracts in Math. 90, Cambridge University Press, Cambridge, 1988. Google Scholar

  • [10]

    Hua L.-K., On the least primitive root of a prime, Bull. Amer. Math. Soc. 48 (1942), 726–730. Google Scholar

  • [11]

    Iwaniec H. and Kowalski E., Analytic Number Theory, American Mathematical Society, Providence, 2004. Google Scholar

  • [12]

    Linnik Y. V., A remark on the least quadratic non-residue (in Russian), C. R. Dokl. Acad. Sci. URSS (N.S.) 36 (1942), 119–120. Google Scholar

  • [13]

    Martin G., The least prime primitive root and the shifted sieve, Acta Arith. 80 (1997), no. 3, 277–288. Google Scholar

  • [14]

    Vinogradov I. M., On the distribution of quadratic residues and nonresidues (in Russian), Ž. Fiz. Mat. Obšč. Univ. Perm 2 (1919), 1–16. Google Scholar

  • [15]

    Vinogradov I. M., On the least primitive root (in Russian), Dokl. Akad. Nauk SSSR 1 (1930), 7–11. Google Scholar

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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