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 $c\in {\mathbb{F}}_q^{\ast }$ and $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}6)$. 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 $n\ge 3$ variables with $c\in {\mathbb{F}}_q^{\ast }$ and $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}6)$, can also be deduced.