The distribution of solutions of the congruence $x_1x_2x_3\dots x_n\equiv c (\mod p)$
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- by Anwar Ayyad PDF
- Proc. Amer. Math. Soc. 127 (1999), 943-950 Request permission
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 $B\gg _{\epsilon }p^{\frac {2}{n}+\epsilon }$.References
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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
- MR Author ID: 609765
- Email: anwar@math.ksu.edu
- Received by editor(s): May 9, 1997
- Communicated by: Dennis A. Hejhal
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 943-950
- MSC (1991): Primary 11D79, 11L40
- DOI: https://doi.org/10.1090/S0002-9939-99-05124-2
- MathSciNet review: 1641700