Brewer sum

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

Definition

The Brewer sum is given by

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

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

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


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