Brewer sum

In mathematics, Brewer sums are finite character sum introduced by Brewer (1961, 1966) related to Jacobsthal sums.

Definition

The Brewer sum is given by

${\displaystyle \Lambda _{n}(a)=\sum _{x{\bmod {p}}}{\binom {D_{n+1}(x,a)}{p}}}$

where D_n is the Dickson polynomial (or "Brewer polynomial") given by

${\displaystyle D_{0}(x,a)=2,\quad D_{1}(x,a)=x,\quad D_{n+1}(x,a)=xD_{n}(x,a)-aD_{n-1}(x,a)}$

and () is the Legendre symbol.

The Brewer sum is zero when n is coprime to q²−1.