The law of quadratic reciprocity, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows to determine the solvability of any quadratic equation in modular arithmetic.
Suppose p and q are two different odd primes. If at least one of them is congruent to 1 modulo 4, then the congruence
Using the Legendre symbol (p/q), these statements may be summarized as
In a book about reciprocity laws published in 2000, Lemmermeyer collects literature citations for 196 different published proofs for the quadratic reciprocity law.
There are cubic, quartic (biquadratic) and other higher reciprocity laws; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws).