 These sums were known and used before Kummer, in the theory of cyclotomy.A Kummer sum is therefore a finite sum

taken over "r" modulo "p", where χ is a Dirichlet character taking values in the cube roots of unity, and where "e"("x") is the exponential function exp(2π"ix").Given "p" of the required form, there are two such characters, together with the trivial character.