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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The distribution of solutions
of the congruence $x_{1}x_{2}x_{3}\dots x_{n}\equiv c\pmod p$


Author: Anwar Ayyad
Journal: Proc. Amer. Math. Soc. 127 (1999), 943-950
MSC (1991): Primary 11D79, 11L40
MathSciNet review: 1641700
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Abstract: For a cube $\mathcal{B}$ of size $B$, we obtain a lower bound on $B$ so that $\mathcal{B}\cap V$ is nonempty, where $V$ is the algebraic subset of $\mathbb{F}_{p}^{n}$ defined by

\begin{equation*}x_{1}x_{2}x_{3}\dots x_{n}\equiv c\pmod p ,\end{equation*}

$n$ a positive integer and $c$ an integer not divisible by $p$. For $n=3$ we obtain that $\mathcal{B}\cap V$ is nonempty if $B\gg p^{\frac{2}{3}}(\log p)^{\frac{2}{3}}$, for $n=4$ we obtain that $\mathcal{B}\cap V$ is nonempty if $B\gg \sqrt {p}\log p$, and for $n\ge 5$ we obtain that $\mathcal{B}\cap V$ is nonempty if $B\gg p^{\frac{1}{4}+\frac{1}{\sqrt {2(n+4)}}}(\log p)^{\frac{3}{2}}$. Using the assumption of the Grand Riemann Hypothesis we obtain $\mathcal{B}\cap V$ is nonempty if $B\gg _{\epsilon }p^{\frac{2}{n}+\epsilon }$.


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Additional Information

Anwar Ayyad
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Address at time of publication: Department of Mathematics, University of Gaza, P.O. Box 1418, Gaza Strip, Via Israel
Email: anwar@math.ksu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05124-2
PII: S 0002-9939(99)05124-2
Keywords: Distribution, congruences, solutions
Received by editor(s): May 9, 1997
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1999 American Mathematical Society