Brewer sum

Short description: Brewet sums are finite numbers introduced by brewer related to Jacobsthal sums

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

Definition

The Brewer sum is given by

[math]\displaystyle{ \Lambda_n(a) = \sum_{x\bmod p}\binom{D_{n+1}(x,a)}{p} }[/math]

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

[math]\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) }[/math]

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 
• 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 
• 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, http://projecteuclid.org/getRecord?id=euclid.ijm/1256048104 
• Lidl, Rudolf; Niederreiter, Harald (1997), Finite fields, Encyclopedia of Mathematics and Its Applications, 20 (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4, https://archive.org/details/finitefields0000lidl_a8r3