# IJPAM: Volume 50, No. 4 (2009)

**ON THE NUMBER OF SOLUTIONS OF THE EQUATION**

IN FINITE FIELDS

IN FINITE FIELDS

Luo Yanmei, Zhao Zhengjun, Cao Xiwang

Department of Mathematics

Nanjing University of Aeronautics and Astronautics

Nanjing, 210016, P.R. CHINA

e-mail: ymluo@nuaa.edu.cn

e-mail: zzj0608115@126.com

e-mail: xwcao@nuaa.edu.cn

Department of Mathematics

Nanjing University of Aeronautics and Astronautics

Nanjing, 210016, P.R. CHINA

e-mail: ymluo@nuaa.edu.cn

e-mail: zzj0608115@126.com

e-mail: xwcao@nuaa.edu.cn

**Abstract.**Let be a finite field with elements, where is an odd prime. Let
denote the number of solutions of the equation

in the finite field . We obtain the explicit formula for , where , , , . We also find explicit formulas for the number of solutions of , where or and .

**Received: **November 11, 2008

**AMS Subject Classification: **11T06, 11T23

**Key Words and Phrases: **finite fields, solutions of equation, quadratic character

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 50

**Issue:** 4