A Gaussian sum is a sum of the form
(1)
where relatively prime integers . The symbol relatively prime integers is often useful, it is not necessary, and Gaussian sums can be written so as to be valid for all integer
If
(2)
(Nagell 1951, p. 178). Gauss showed that
(3)
for odd
(4)
(Nagell 1951, p. 177).
For parity (i.e., one is even and the other is odd ), Schaar's identity states
(5)
Such sums are important in the theory of quadratic residues .
See also Kloosterman's Sum ,
Quadratic Residue ,
Schaar's Identity ,
Singular Series Explore with Wolfram|Alpha References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. Evans, R. and Berndt, B. "The Determination of Gauss Sums." Bull. Amer. Math. Soc. 5 , 107-129, 1981. Katz, N. M. Gauss Sums, Kloosterman Sums, and Monodromy Groups. Malyšev, A. V. "Gauss and Kloosterman Sums." Dokl. Akad. Nauk SSSR 133 , 1017-1020, 1960. English translation in Soviet Math. Dokl. 1 , 928-932, 1960. Nagell, T. "The Gaussian Sums." §53 in Introduction to Number Theory. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Referenced on Wolfram|Alpha Gaussian Sum Cite this as: Weisstein, Eric W. "Gaussian Sum." From MathWorld https://mathworld.wolfram.com/GaussianSum.html
Subject classifications 
and 
are relatively prime integers. The symbol 
is sometimes used instead of 
. Although the restriction to relatively prime integers is often useful, it is not necessary, and Gaussian sums can be written so as to be valid for all integer 
(Borwein and Borwein 1987, pp. 83 and 86). 
, then 
. Written explicitly 
and 
of opposite parity (i.e., one is even and the other is odd), Schaar's identity states 
