The quadratic reciprocity theorem states that if and are distinct odd primes, then the congruences

are both solvable or both unsolvable unless both and leave the remainder 3 when divided by 4 (in which case one of the congruences is solvable and the other is not). Written symbolically,

where

is known as a Legendre symbol.

Gauss called this result the "aureum theorema" (golden theorem).

Euler stated the theorem in 1783 without proof. Legendre was the first to publish a proof, but it was fallacious. In 1796, Gauss became the first to publish a correct proof (Nagell 1951, p. 144). The quadratic reciprocity theorem was Gauss's favorite theorem from number theory, and he devised no fewer than eight different proofs of it over his lifetime.

The genus theorem states that the Diophantine equation

can be solved for a prime iff or .

See also

Genus Theorem, Jacobi Symbol, Kronecker Symbol, Legendre Symbol, Quadratic, Quadratic Residue, Reciprocity Theorem Explore this topic in the MathWorld classroom

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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.Jones, G. A. and Jones, J. M. "Quadratic Reciprocity." §7.4 in Elementary Number Theory. Berlin, Germany: Springer-Verlag, pp. 130-135, 1998.Nagell, T. "The Quadratic Reciprocity Law." §41 in Introduction to Number Theory. New York: Wiley, pp. 141-145, 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.

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Quadratic Reciprocity Theorem

Cite this as:

Weisstein, Eric W. "Quadratic Reciprocity Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuadraticReciprocityTheorem.html

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