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The congruence $ ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p$


Authors: Anwar Ayyad and Todd Cochrane
Journal: Proc. Amer. Math. Soc. 145 (2017), 467-477
MSC (2010): Primary 11A07, 11L05, 11L40, 11D79
DOI: https://doi.org/10.1090/proc/13429
Published electronically: October 31, 2016
MathSciNet review: 3577853
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Abstract: For prime $ p$ and integers $ a,b,c$ with $ p \nmid ab$, we obtain solutions of the congruence

$\displaystyle ax_1x_2 \cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k}\equiv c \pmod p$    

in a cube $ \mathcal B$ with edge length $ B$. For a cube in general position, we show that if $ p \nmid abc$ and $ k \ge 5$, then the congruence above has a solution in any cube with edge length $ B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }.$ Estimates are given for the case $ p\vert c$ as well, and improvements are given for small $ k$. For cubes cornered at the origin, $ 1 \le x_i \le B$ for all $ i$, we obtain a solution provided only that $ B\gg p^{\frac 3{2k} + O\left (\frac k{\log \log p}\right )}.$ Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.

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Additional Information

Anwar Ayyad
Affiliation: Department of Mathematics, Al Azhar University, P.O. Box 1277, Gaza Strip, Palestine
Email: anwarayyad@yahoo.com

Todd Cochrane
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: cochrane@math.ksu.edu

DOI: https://doi.org/10.1090/proc/13429
Keywords: Modular hyperbolas, multiplicative congruences, sum-product sets
Received by editor(s): October 31, 2015
Published electronically: October 31, 2016
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society