On the number of solutions of two-variable diagonal sextic equations over finite fields

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.