Vol. 25, No. 3, 1968

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Theorems on Brewer sums

Stanley F. Robinson

Vol. 25 (1968), No. 3, 587–596

Let V m(x,Q) be the polynomial determined by the recurrence relation

Vm+2 (x,Q) = x ⋅Vm+1(x,Q) − Q⋅Vm (x,Q)

(m = 1,2,), Q an integer, with V 1(x,Q) = x and V 2(x,Q) = x2 2Q. In a recent paper, B. W. Brewer has defined the sum

Λm (Q ) =   χ(Vm (x,Q ))

where χ(s) denotes the Legendre symbol (s∕p) with p and odd prime.

The purpose of this paper is to consider the evaluation of Λ2n(Q) when n is odd. The principle result obtained is the expression of Λ2n(Q) as the sum of χ(Q) Λn(1) and one half the character sum ψ2e(1). ψ2e(1) can in turn be expressed in terms of the Gaussian cyclotomic numbers (i,j). The values of Λ6(Q) and Λ10(Q) follow immediately from this result utilizing values for Λ3(1) = Λ3 and Λ5(1) = Λ5 computed by B. W. Brewer and A. L. Whiteman.

Mathematical Subject Classification
Primary: 10.41
Received: 8 August 1966
Revised: 1 November 1967
Published: 1 June 1968
Stanley F. Robinson