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Mat. Sb. (N.S.), 1979, Volume 109(151), Number 2(6), Pages 171–187 (Mi msb2363)  

This article is cited in 8 scientific papers (total in 8 papers)

On the number of solutions of an $n$th degree congruence with one unknown

S. V. Konyagin


Abstract: We show that the number of solutions to the congruence $f(x)\equiv 0\pmod m$, where $f(x)$ is a polynomial of degree $n\geqslant2$ whose coefficients have greatest common divisor relatively prime to $m$, does not exceed $(n/e+O(\ln^2 n))m^{1-1/n}$, where $n/e+O(\ln^2n)$ cannot be replaced by $n/e$.
Bibliography: 5 titles.

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English version:
Mathematics of the USSR-Sbornik, 1980, 37:2, 151–166

Bibliographic databases:

Document Type: Article
UDC: 511.22
MSC: Primary 10A10, 10G05; Secondary 12B05
Received: 11.07.1978

Citation: S. V. Konyagin, “On the number of solutions of an $n$th degree congruence with one unknown”, Mat. Sb. (N.S.), 109(151):2(6) (1979), 171–187; Math. USSR-Sb., 37:2 (1980), 151–166

Citation in format AMSBIB
\Bibitem{Kon79}
\by S.~V.~Konyagin
\paper On the number of solutions of an $n$th degree congruence with one unknown
\jour Mat. Sb. (N.S.)
\yr 1979
\vol 109(151)
\issue 2(6)
\pages 171--187
\mathnet{http://mi.mathnet.ru/msb2363}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=542556}
\zmath{https://zbmath.org/?q=an:0447.10005|0406.10003}
\transl
\jour Math. USSR-Sb.
\yr 1980
\vol 37
\issue 2
\pages 151--166
\crossref{https://doi.org/10.1070/SM1980v037n02ABEH001947}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980KQ02900001}


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  • http://mi.mathnet.ru/eng/msb/v151/i2/p171

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    Erratum

    This publication is cited in the following articles:
    1. D. A. Mit'kin, “On estimates and asymptotic formulas for rational trigonometric sums that are almost complete”, Math. USSR-Sb., 50:2 (1985), 513–532  mathnet  crossref  mathscinet  zmath
    2. Shparlinskii I., “Polynomial Congruences”, Acta Arith., 58:2 (1991), 153–156  mathscinet  isi
    3. S. V. Konyagin, T. Steger, “On polynomial congruences”, Math. Notes, 55:6 (1994), 596–600  mathnet  crossref  mathscinet  zmath  isi
    4. Coppersmith D., Shparlinski I., “On Polynomial Approximation of the Discrete Logarithm and the Diffie-Hellman Mapping”, J. Cryptology, 13:3 (2000), 339–360  crossref  mathscinet  zmath  isi
    5. Blackburn S., Gomez-Perez D., Gutierrez J., Shparlinski I., “Predicting Nonlinear Pseudorandom Number Generators”, Math. Comput., 74:251 (2005), 1471–1494  mathscinet  zmath  adsnasa  isi
    6. Luca F., Shparlinski I.E., “Pseudoprime Values of the Fibonacci Sequence, Polynomials and the Euler Function”, Indag. Math.-New Ser., 17:4 (2006), 611–625  crossref  mathscinet  zmath  isi
    7. Blackburn S.R., Gomez-Perez D., Gutierrez J., Shparlinski I.E., “Reconstructing Noisy Polynomial Evaluation in Residue Rings”, J. Algorithms, 61:2 (2006), 47–59  crossref  mathscinet  zmath  isi
    8. Chen Zh., Winterhof A., “On the Distribution of Pseudorandom Numbers and Vectors Derived From Euler-Fermat Quotients”, Int. J. Number Theory, 8:3 (2012), 631–641  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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