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Quadratic reciprocity law

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The quadratic reciprocity law concerns congruences modulo odd primes. The solubility of and depends on whether both of the primes are congruent to 3 modulo 4.

Theorem. Given distinct odd primes and ,

where is the Legendre symbol.


For example, both 13 and 17 are congruent to 1 modulo 4. We see that and . These two primes "reciprocate" each other as quadratic residues.

Compare 11 and 19. The quadratic reciprocity law tells us one is quadratic residue of the other but not vice-versa. Indeed , but has no solutions. These two primes do not "reciprocate."

Gauss's first proof of April 8, 1796 is now considered inelegant by some. Soon after that same year, on June 27, Gauss came up with a proof using quadratic forms.