Brewer sum

In mathematics, Brewer sums are finite character sums introduced by Brewer related to Jacobsthal sums.

Let ${\displaystyle a\in \mathbb {Z} /q\mathbb {Z} }$, with ${\displaystyle q}$ a power of some prime ${\displaystyle p}$. The Brewer sum is defined as

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

where ${\displaystyle ({\frac {\;}{\;}})}$ is the Legendre symbol and ${\displaystyle D_{n}}$ is the Dickson polynomial given by

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

The Brewer sum is zero when ${\displaystyle n}$ is coprime to ${\displaystyle q^{2}-1}$.

Notes

References

This number theory–related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e