  Using properties of Gauss and Jacobi sums, we derive explicit formulas for
the number of solutions to a diagonal equation of the form
$x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm
3\pmod{8}$. All of the evaluations are effected in terms of parameters
occurring in quadratic partitions of some powers of $p$.
