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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

${\displaystyle \phi _{n}(a)=\sum _{m{\bmod {p}}}\left({\dfrac {m(m^{n}+a)}{p}}\right)}$

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

References

• 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 0537798, Zbl 0393.12029
• 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, JFM 38.0238.03

Further reading

• Storer, Thomas (1967), Cyclotomy and difference sets, Chicago: Markham Publishing Company, Zbl 0157.03301

• Number theory