Quadratic reciprocity
From Number
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) |
