Quadratic Reciprocity Theorem

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

(1)

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,

(2)

where

(3)

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

(4)

can be solved for a prime iff or .

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