Editing Quadratic reciprocity (section)

===Patterns among quadratic residues===
Let ''p'' be an odd prime. A number modulo ''p'' is a [[quadratic residue]] whenever it is congruent to a square (mod ''p''); otherwise it is a quadratic non-residue. ("Quadratic" can be dropped if it is clear from the context.) Here we exclude zero as a special case. Then as a consequence of the fact that the multiplicative group of a [[finite field]] of order ''p'' is cyclic of order ''p-1'', the following statements hold:

*There are an equal number of quadratic residues and non-residues; and
*The product of two quadratic residues is a residue, the product of a residue and a non-residue is a non-residue, and the product of two non-residues is a residue.

For the avoidance of doubt, these statements do ''not'' hold if the modulus is not prime. 
For example, there are only 3 quadratic residues (1, 4 and 9) in the multiplicative group modulo 15. 
Moreover, although 7 and 8 are quadratic non-residues, their product 7x8 = 11 is also a quadratic non-residue, in contrast to the prime case.

Quadratic residues appear as entries in the following table, indexed by the row number as modulus and column number as root:

{| class="wikitable" style="text-align:right;" cellpadding="2"
|+ Squares mod primes
|-
! ''n'' 
| 1||2|| 3|| 4||5|| 6|| 7|| 8 ||9 ||10|| 11|| 12|| 13|| 14 ||15|| 16|| 17|| 18 ||19|| 20|| 21|| 22|| 23|| 24 ||25
|-
! ''n''<sup>2</sup> 
|1 
|4
|9
|16
|25
|36
|49
|64 
|81 
|100|| 121|| 144|| 169|| 196 ||225|| 256|| 289|| 324 ||361 || 400|| 441|| 484|| 529|| 576 ||625
|-
! mod 3 
| 1 || 1 || 0
| 1 || 1 || 0
| 1 || 1 || 0
| 1 || 1 || 0
| 1 || 1 || 0
| 1 || 1 || 0
| 1 || 1 || 0
| 1 || 1 || 0
| 1 
|-
! mod 5 
| 1 || 4 || 4 || 1 || 0
| 1 || 4 || 4 || 1 || 0
| 1 || 4 || 4 || 1 || 0
| 1 || 4 || 4 || 1 || 0
| 1 || 4 || 4 || 1 || 0
|-
! mod 7 
| 1 || 4 || 2 || 2 || 4 || 1 || 0 
| 1 || 4 || 2 || 2 || 4 || 1 || 0 
| 1 || 4 || 2 || 2 || 4 || 1 || 0 
| 1 || 4 || 2 || 2 
|-
! mod 11 
| 1 || 4 || 9 || 5 || 3 || 3 || 5 || 9 || 4 || 1 || 0 
| 1 || 4 || 9 || 5 || 3 || 3 || 5 || 9 || 4 || 1 || 0 
| 1 || 4 || 9 
|-
! mod 13 
| 1 || 4 || 9 || 3 || 12 || 10 || 10 || 12 || 3 || 9 || 4 || 1 || 0 
| 1 || 4 || 9 || 3 || 12 || 10 || 10 || 12 || 3 || 9 || 4 || 1 
|-
! mod 17 
| 1 || 4 || 9 || 16 || 8 || 2 || 15 || 13 || 13 || 15 || 2 || 8 || 16 || 9 || 4 || 1 || 0 
| 1 || 4 || 9 || 16 || 8 || 2 || 15 || 13 
|-
! mod 19 
| 1 || 4 || 9 || 16 || 6 || 17 || 11 || 7 || 5 || 5 || 7 || 11 || 17 || 6 || 16 || 9 || 4 || 1 || 0
| 1 || 4 || 9 || 16 || 6 || 17 
|-
! mod 23 
| 1 || 4 || 9 || 16 || 2 || 13 || 3 || 18 || 12 || 8 || 6 || 6 || 8 || 12 ||18 || 3 || 13 || 2 || 16 || 9 || 4 || 1 || 0 
| 1 || 4 
|-
! mod 29 
| 1 ||4|| 9|| 16|| 25|| 7|| 20|| 6 ||23 ||13|| 5|| 28|| 24|| 22 ||22|| 24|| 28|| 5 ||13 || 23|| 6|| 20|| 7|| 25 ||16
|-
! mod 31 
| 1 ||4|| 9|| 16||25|| 5|| 18|| 2 ||19 ||7|| 28|| 20|| 14|| 10 ||8|| 8|| 10|| 14 ||20 || 28|| 7|| 19|| 2|| 18 ||5
|-
! mod 37 
| 1 ||4|| 9|| 16||25|| 36|| 12|| 27 ||7 ||26||10|| 33|| 21|| 11 ||3|| 34|| 30|| 28 ||28 || 30|| 34|| 3|| 11|| 21 ||33
|- 
! mod 41 
| 1 ||4|| 9|| 16||25|| 36|| 8|| 23 ||40 ||18||39||21|| 5|| 32 ||20|| 10|| 2|| 37 ||33 || 31|| 31|| 33|| 37|| 2 ||10
|-
! mod 43 
| 1 ||4|| 9|| 16||25|| 36|| 6|| 21 ||38 ||14|| 35|| 15|| 40|| 24 ||10|| 41|| 31|| 23 ||17 || 13|| 11 || 11|| 13|| 17 ||23
|-
! mod 47 
| 1 ||4|| 9|| 16||25|| 36|| 2|| 17 ||34 ||6|| 27|| 3|| 28|| 8 ||37||21||7||42||32||24||18|| 14|| 12 || 12 ||14
|}

This table is complete for odd primes less than 50. To check whether a number ''m'' is a quadratic residue mod one of these primes ''p'', find ''a'' ≡ ''m'' (mod ''p'') and 0 ≤ ''a'' < ''p''. If ''a'' is in row ''p'', then ''m'' is a residue (mod ''p''); if ''a'' is not in row ''p'' of the table, then ''m'' is a nonresidue (mod ''p'').

The quadratic reciprocity law is the statement that certain patterns found in the table are true in general.