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Brewer sum

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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

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 q−1.

References


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