Vipul: Created page with '==Statement== If $p,q$ are distinct odd prime numbers, then: $\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}}$ where …'

2010-05-29T22:14:34Z

Created page with '==Statement== If <math>p,q</math> are distinct odd prime numbers, then: <math>\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}}</math> where …'

New page

==Statement==

If <math>p,q</math> are distinct odd [[prime number]]s, then:

<math>\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}}</math>

where <math>\left(\frac{p}{q}\right)</math> and <math>\left(\frac{q}{p}\right)</math> are the respective [[fact about::Legendre symbol]]s: a Legendre symbol takes the value <math>+1</math> if the top value is a [[fact about::quadratic residue]] modulo the bottom value, and <math>-1</math> if the top value is a [[fact about::quadratic nonresidue]] modulo the bottom value.

The statement can also be captured using the following cases for the residue classes of <matH>p</math> and <math>q</math> modulo <math>4</math>:

{| class="wikitable" border="1"
! Congruence class of <math>p</math> mod <math>4</math> !! Congruence class of <math>q</math> mod <math>4</math> !! Both quadratic residues mod each other? !! Both quadratic nonresidues mod each other? !! <math>p</math> quadratic nonresidue mod <math>q</math>, <math>q</math> quadratic residue mod <math>p</math>? !! <math>p</math> quadratic residue mod <math>q</math>, <math>q</math> quadratic nonresidue mod <math>p</math>?
|-
| 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)
|}

Quadratic reciprocity - Revision history

Vipul: Created page with '==Statement== If p,q are distinct odd prime numbers, then: \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}} where …'

Vipul: Created page with '==Statement== If $p,q$ are distinct odd prime numbers, then: $\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}{4}}$ where …'