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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)