In mathematics, Brewer sums are finite character sum introduced by Brewer (1961, 1966) related to Jacobsthal sums.
Definition The Brewer sum is given by
Λ n ( a ) = ∑ x mod p ( D n + 1 ( x , a ) p ) {\displaystyle \Lambda _{n}(a)=\sum _{x{\bmod {p}}}{\binom {D_{n+1}(x,a)}{p}}}
where Dn is the Dickson polynomial (or "Brewer polynomial") given by
D 0 ( x , a ) = 2 , D 1 ( x , a ) = x , D n + 1 ( x , a ) = x D n ( x , a ) − a D n − 1 ( x , a ) {\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.
References
Brewer, B. W. (1961), "On certain character sums", Transactions of the American Mathematical Society, 99 (2): 241–245, doi:10.2307/1993392, ISSN 0002-9947, JSTOR 1993392, MR 0120202, Zbl 0103.03205 Brewer, B. W. (1966), "On primes of the form u²+5v²", Proceedings of the American Mathematical Society, 17 (2): 502–509, doi:10.2307/2035200, ISSN 0002-9939, JSTOR 2035200, MR 0188171, Zbl 0147.29801 Berndt, Bruce C.; Evans, Ronald J. (1979), "Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer", Illinois Journal of Mathematics, 23 (3): 374–437, doi:10.1215/ijm/1256048104, ISSN 0019-2082, MR 0537798, Zbl 0393.12029 Lidl, Rudolf; Niederreiter, Harald (1997), Finite fields, Encyclopedia of Mathematics and Its Applications, vol. 20 (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069