Quadratic reciprocity

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Statement

If are distinct odd prime numbers, then:

where and are the respective Legendre symbols: a Legendre symbol takes the value if the top value is a quadratic residue modulo the bottom value, and if the top value is a quadratic nonresidue modulo the bottom value.

The statement can also be captured using the following cases for the residue classes of and modulo :

Congruence class of mod Congruence class of mod Both quadratic residues mod each other? Both quadratic nonresidues mod each other? quadratic nonresidue mod , quadratic residue mod ? quadratic residue mod , quadratic nonresidue mod ?
1 1 Possible (example: 5, 29) Possible (example: 5, 13) Impossible Impossible
1 -1 Possible (example: 5, 19) Possible (example: 5, 7) Impossible Impossible
-1 1 Possible (example: 19, 5) Possible (example: 7, 5) Impossible Impossible
-1 -1 Impossible Impossible Possible (example: 3,7) Possible (example: 7, 3)