On Diagonal Equations over Finite Fields,

In this paper, we obtain a sufficient condition for the diagonal equation to have only the trivial solution over finite fields. This result improves a theorem of Sun (J. Sichuan Normal Univ. Nat. Sci. Ed.26 (1989), 55–59) greatly and proves that the conjecture posed by Powell (J. Number Theory18 (1984), 34–40) holds for general n N as well.