@article{Ge2024, 
author = {Wenxu Ge and Weiping Li and Tianze Wang},
title = {A note on some diagonal cubic equations over finite fields},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {8},
pages = {21656-21671},
keywords = {Gauss sum, finite field, diagonal cubic equation},
url = {https://www.sciopen.com/article/10.3934/math.20241053},
doi = {10.3934/math.20241053},
abstract = {Let a prime  p≡1(mod3) and  z be non-cubic in  Fp. Gauss proved that the number of solutions of equation   x13+x23+zx33=0in  Fp was  p2+12(p−1)(9d−c), where  c was uniquely determined and  d, except for the sign, was defined by   4p=c2+27d2,c≡1(mod3).In 1978, Chowla, Cowles, and Cowles determined the sign of  d for the case of 2 being non-cubic in  Fp. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field  Fq with  q=pk,  p≡1(mod3), and determined the sign of  d for the case of 3 being non-cubic.}
}