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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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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
• Email
• Other articles by this author:
/ Moubariz Z. Garaev
• Corresponding author
• Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, Mexico
• Email
• Other articles by this author:
/ 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 ${p}_{1},\mathrm{\dots },{p}_{r}$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer n which is a simultaneous ${p}_{1},\mathrm{\dots },{p}_{r}$-power nonresidue modulo p satisfies

$n<{p}^{1/4-{c}_{r}+o\left(1\right)}\mathit{ }\left(p\to \mathrm{\infty }\right)$

for some positive ${c}_{r}$ satisfying ${c}_{r}={e}^{-\left(1+o\left(1\right)\right)r}$ as $r\to \mathrm{\infty }$.

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

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,

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