Let $ p $ be a prime, $ k $ a positive integer, $ q = p^k $, and $ \mathbb{F}_q $ be the finite field with $ q $ elements. In this paper, by using the Jacobi sums, we give an explicit formula for the number of solutions of the two-variable diagonal sextic equations $ x_1^6+x_2^6 = c $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $ and $ p\equiv1({\rm{mod}} \ 6) $. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the diagonal sextic equations $ x_1^6+x_2^6+\cdots+x_n^6 = c $ of $ n\geq3 $ variables with $ c\in\mathbb{F}_q^* $ and $ p\equiv1({\rm{mod}} \ 6) $, can also be deduced.
Citation: Shuangnian Hu, Rongquan Feng. On the number of solutions of two-variable diagonal sextic equations over finite fields[J]. AIMS Mathematics, 2022, 7(6): 10554-10563. doi: 10.3934/math.2022588
Let $ p $ be a prime, $ k $ a positive integer, $ q = p^k $, and $ \mathbb{F}_q $ be the finite field with $ q $ elements. In this paper, by using the Jacobi sums, we give an explicit formula for the number of solutions of the two-variable diagonal sextic equations $ x_1^6+x_2^6 = c $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $ and $ p\equiv1({\rm{mod}} \ 6) $. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the diagonal sextic equations $ x_1^6+x_2^6+\cdots+x_n^6 = c $ of $ n\geq3 $ variables with $ c\in\mathbb{F}_q^* $ and $ p\equiv1({\rm{mod}} \ 6) $, can also be deduced.
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Shuangnian Hu, Rongquan Feng. On the number of solutions of two-variable diagonal sextic equations over finite fields[J]. AIMS Mathematics, 2022, 7(6): 10554-10563. doi: 10.3934/math.2022588
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