Also called the Aureum Theorema (Golden Theorem) by Gauß. If and are distinct Odd Primes, then the Congruences

where

is known as a Legendre Symbol. Legendre was the first to publish a proof, but it was fallacious. Gauß was the first to publish a correct proof. The quadratic reciprocity theorem was Gauss's favorite theorem from Number Theory, and he devised many proofs of it over his lifetime.

**References**

Courant, R. and Robbins, H. *What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, p. 39, 1996.

Ireland, K. and Rosen, M. ``Quadratic Reciprocity.'' Ch. 5 in *A Classical Introduction to Modern Number Theory, 2nd ed.*
New York: Springer-Verlag, pp. 50-65, 1990.

Nagell, T. ``Theory of Quadratic Residues.'' Ch. 4 in *Introduction to Number Theory.* New York: Wiley, 1951.

Riesel, H. ``The Law of Quadratic Reciprocity.'' *Prime Numbers and Computer Methods for Factorization, 2nd ed.*
Boston, MA: Birkhäuser, pp. 279-281, 1994.

Shanks, D. *Solved and Unsolved Problems in Number Theory, 4th ed.* New York: Chelsea, pp. 42-49, 1993.

© 1996-9

1999-05-25