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The distribution of solutions of the congruence

Authors:	Anwar Ayyad

Institution:	Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Abstract:	For a cube $\mathcal{B}$ of size , we obtain a lower bound on so that $\mathcal{B}\cap V$ is nonempty, where is the algebraic subset of $\mathbb{F}_{p}^{n}$ defined by $\begin{equation}x_{1}x_{2}x_{3}\dots x_{n}\equiv c\pmod p ,\end{equation}$ a positive integer and an integer not divisible by . For we obtain that $\mathcal{B}\cap V$ is nonempty if $B\gg p^{\frac{2}{3}}(\log p)^{\frac{2}{3}}$ , for 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 $B\gg _{\epsilon }p^{\frac{2}{n}+\epsilon }$ .

Keywords:	Distribution congruences solutions

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