Gaussian Sum

 (1)

where and are Relatively Prime Integers. If , then
 (2)

Gauß showed
 (3)

for Odd . A more general result was obtained by Schaar. For and of opposite Parity (i.e., one is Even and the other is Odd), Schaar's Identity states
 (4)

Such sums are important in the theory of Quadratic Residues.

References

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. Princeton, NJ: Princeton University Press, 1987.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 132-134, 1994.