Jacobsthal sum

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In mathematics, Jacobsthal sums are finite sums of Legendre symbols related to Gauss sums. They were introduced by Jacobsthal (1907).

Definition

The Jacobsthal sum is given by

$\phi _{n}(a)=\sum _{{m{\bmod p}}}{\binom {m(m^{n}+a)}{p}}$

where p is prime and () is the Legendre symbol.

Berndt, Bruce C.; Evans, Ronald J. (1979), "Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer", Illinois Journal of Mathematics 23 (3): 374–437, ISSN 0019-2082, MR 537798
Jacobsthal, E. (1907), "Über die Darstellung der Primzahlen der Form 4n + 1 als Summe zweier Quadrate", Journal für die reine und angewandte Mathematik: 238–245, ISSN 0075-4102

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