- User:James in dc/sandbox/Divisor theory
- User:James in dc/sandbox/Jacobi symbol
- User:James in dc/sandbox/Quadratic Reciprocity
table
editAnother way to organize the data is to see which primes are residues mod which other primes, as illustrated in the following table. The entry in row p column q is R if q is a quadratic residue (mod p); if it is a nonresidue the entry is N.
If the row, or the column, or both, are ≡ 1 (mod 4) the entry is blue or green; if both row and column are ≡ 3 (mod 4), it is yellow or orange.
The blue and green entries are symmetric around the diagonal: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is R (resp N).
The yellow and orange ones, on the other hand, are antisymmetric: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is N (resp R).
The reciprocity law states that these patterns hold for all p and q.
| R | q is a residue (mod p) | q ≡ 1 (mod 4) or p ≡ 1 (mod 4) (or both) (reciprocating) |
| N | q is a nonresidue (mod p) | |
| R | q is a residue (mod p) | q ≡ p ≡ 3 (mod 4) (non-reciprocating) |
| N | q is a nonresidue (mod p) |
| q | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | ||
| p | 3 | N | R | N | R | N | R | N | N | R | R | N | R | N | N | N | R | R | N | R | R | N | N | R | |
| 5 | N | N | R | N | N | R | N | R | R | N | R | N | N | N | R | R | N | R | N | R | N | R | N | ||
| 7 | N | N | R | N | N | N | R | R | N | R | N | R | N | R | N | N | R | R | N | R | N | N | N | ||
| 11 | R | R | N | N | N | N | R | N | R | R | N | N | R | R | R | N | R | R | N | N | N | R | R | ||
| 13 | R | N | N | N | R | N | R | R | N | N | N | R | N | R | N | R | N | N | N | R | N | N | N | ||
| 17 | N | N | N | N | R | R | N | N | N | N | N | R | R | R | R | N | R | N | N | N | R | R | N | ||
| 19 | N | R | R | R | N | R | R | N | N | N | N | R | R | N | N | R | N | N | R | N | R | N | N | ||
| 23 | R | N | N | N | R | N | N | R | R | N | R | N | R | N | R | N | N | R | R | N | N | N | N | ||
| 29 | N | R | R | N | R | N | N | R | N | N | N | N | N | R | R | N | R | R | N | N | R | N | N | ||
| 31 | N | R | R | N | N | N | R | N | N | N | R | N | R | N | R | N | R | R | N | N | N | N | R | ||
| 37 | R | N | R | R | N | N | N | N | N | N | R | N | R | R | N | N | R | R | R | N | R | N | N | ||
| 41 | N | R | N | N | N | N | N | R | N | R | R | R | N | N | R | R | N | N | R | N | R | N | N | ||
| 43 | N | N | N | R | R | R | N | R | N | R | N | R | R | R | R | N | R | N | N | R | R | N | R | ||
| 47 | R | N | R | N | N | R | N | N | N | N | R | N | N | R | R | R | N | R | N | R | R | R | R | ||
| 53 | N | N | R | R | R | R | N | N | R | N | R | N | R | R | R | N | N | N | N | N | N | R | R | ||
| 59 | R | R | R | N | N | R | R | N | R | N | N | R | N | N | R | N | N | R | N | R | N | N | N | ||
| 61 | R | R | N | N | R | N | R | N | N | N | N | R | N | R | N | N | N | N | R | N | R | N | R | ||
| 67 | N | N | N | N | N | R | R | R | R | N | R | N | N | R | N | R | N | R | R | N | R | R | N | ||
| 71 | R | R | N | N | N | N | R | N | R | N | R | N | R | N | N | N | N | N | R | R | R | R | N | ||
| 73 | R | N | N | N | N | N | R | R | N | N | R | R | N | N | N | N | R | R | R | R | N | R | R | ||
| 79 | N | R | N | R | R | N | R | R | N | R | N | N | N | N | N | N | N | R | N | R | R | R | R | ||
| 83 | R | N | R | R | N | R | N | R | R | R | R | R | N | N | N | R | R | N | N | N | N | N | N | ||
| 89 | N | R | N | R | N | R | N | N | N | N | N | N | N | R | R | N | N | R | R | R | R | N | R | ||
| 97 | R | N | N | R | N | N | N | N | N | R | N | N | R | R | R | N | R | N | N | R | R | N | R | ||
new table
edit| q | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 83 | 79 | 71 | 67 | 59 | 47 | 43 | 31 | 23 | 19 | 11 | 7 | 3 | 5 | 13 | 17 | 29 | 37 | 41 | 53 | 61 | 73 | 89 | 97 | ||
| p | 3 | N | R | N | R | N | N | R | R | N | R | N | R | N | R | N | N | R | N | N | R | R | N | R | |
| 5 | N | R | R | N | R | N | N | R | N | R | R | N | N | N | N | R | N | R | N | R | N | R | N | ||
| 7 | N | R | R | R | N | N | R | N | R | N | R | N | N | N | N | R | R | N | R | N | N | N | N | ||
| 11 | N | N | R | R | R | R | N | R | R | N | N | R | R | N | N | N | R | N | R | N | N | R | R | ||
| 13 | N | R | N | N | N | N | R | N | R | N | N | N | R | N | R | R | N | N | R | R | N | N | N | ||
| 17 | R | N | N | R | R | R | R | N | N | R | N | N | N | N | R | N | N | N | R | N | N | R | N | ||
| 19 | R | N | N | N | N | R | R | N | R | R | R | N | R | N | R | N | N | N | N | R | R | N | N | ||
| 23 | N | N | R | N | R | R | N | R | N | N | N | R | N | R | N | R | N | R | N | N | R | N | N | ||
| 29 | R | N | R | R | R | N | N | N | R | N | N | R | N | R | R | N | N | N | R | N | N | N | N | ||
| 31 | N | N | R | R | R | R | N | N | R | N | R | N | R | N | N | N | N | R | N | N | N | N | R | ||
| 37 | R | N | R | R | N | R | N | N | N | N | R | R | R | N | N | N | N | R | R | N | R | N | N | ||
| 41 | R | N | N | N | R | N | R | R | R | N | N | N | N | R | N | N | N | R | N | R | R | N | N | ||
| 43 | R | R | N | R | R | R | R | R | N | R | N | N | N | R | R | N | N | R | R | N | N | N | R | ||
| 47 | R | R | R | N | R | N | N | N | N | N | R | R | N | N | R | N | R | N | R | R | N | R | R | ||
| 53 | N | N | N | N | R | R | R | N | N | N | R | R | N | N | R | R | R | R | N | N | N | R | R | ||
| 59 | N | R | R | N | N | N | N | N | R | N | R | R | R | N | R | R | N | R | R | N | N | N | N | ||
| 61 | R | N | N | N | N | R | N | N | N | R | N | N | R | R | R | N | N | N | R | N | R | N | R | ||
| 67 | R | N | R | R | R | N | N | R | R | N | N | N | N | N | R | R | R | N | N | N | R | R | N | ||
| 71 | R | R | N | N | N | R | N | N | R | N | N | R | R | N | N | R | R | N | N | N | R | R | N | ||
| 73 | N | R | R | R | N | N | N | N | R | R | N | N | R | N | N | N | N | R | R | N | R | R | R | ||
| 79 | R | N | R | N | N | N | R | R | R | R | N | N | R | R | N | N | N | N | N | N | R | R | R | ||
| 83 | N | N | N | R | N | N | R | R | N | R | R | R | N | N | R | R | R | R | N | R | N | N | N | ||
| 89 | N | R | R | R | N | R | N | N | N | N | R | N | N | R | N | R | N | N | N | R | N | R | R | ||
| 97 | N | R | N | N | N | R | R | R | N | N | R | N | R | N | N | N | N | N | N | R | R | R | R | ||
new new table
edit| q | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 83 | 79 | 71 | 67 | 59 | 47 | 43 | 31 | 23 | 19 | 11 | 7 | 3 | 5 | 13 | 17 | 29 | 37 | 41 | 53 | 61 | 73 | 89 | 97 | ||
| p | 83 | N | N | N | R | N | N | R | R | N | R | R | R | N | N | R | R | R | R | N | R | N | N | N | |
| 79 | R | N | R | N | N | N | R | R | R | R | N | N | R | R | N | N | N | N | N | N | R | R | R | ||
| 71 | R | R | N | N | N | R | N | N | R | N | N | R | R | N | N | R | R | N | N | N | R | R | N | ||
| 67 | R | N | R | R | R | N | N | R | R | N | N | N | N | N | R | R | R | N | N | N | R | R | N | ||
| 59 | N | R | R | N | N | N | N | N | R | N | R | R | R | N | R | R | N | R | R | N | N | N | N | ||
| 47 | R | R | R | N | R | N | N | N | N | N | R | R | N | N | R | N | R | N | R | R | N | R | R | ||
| 43 | R | R | N | R | R | R | R | R | N | R | N | N | N | R | R | N | N | R | R | N | N | N | R | ||
| 31 | N | N | R | R | R | R | N | N | R | N | R | N | R | N | N | N | N | R | N | N | N | N | R | ||
| 23 | N | N | R | N | R | R | N | R | N | N | N | R | N | R | N | R | N | R | N | N | R | N | N | ||
| 19 | R | N | N | N | N | R | R | N | R | R | R | N | R | N | R | N | N | N | N | R | R | N | N | ||
| 11 | N | N | R | R | R | R | N | R | R | N | N | R | R | N | N | N | R | N | R | N | N | R | R | ||
| 7 | N | R | R | R | N | N | R | N | R | N | R | N | N | N | N | R | R | N | R | N | N | N | N | ||
| 3 | N | R | N | R | N | N | R | R | N | R | N | R | N | R | N | N | R | N | N | R | R | N | R | ||
| 5 | N | R | R | N | R | N | N | R | N | R | R | N | N | N | N | R | N | R | N | R | N | R | N | ||
| 13 | N | R | N | N | N | N | R | N | R | N | N | N | R | N | R | R | N | N | R | R | N | N | N | ||
| 17 | R | N | N | R | R | R | R | N | N | R | N | N | N | N | R | N | N | N | R | N | N | R | N | ||
| 29 | R | N | R | R | R | N | N | N | R | N | N | R | N | R | R | N | N | N | R | N | N | N | N | ||
| 37 | R | N | R | R | N | R | N | N | N | N | R | R | R | N | N | N | N | R | R | N | R | N | N | ||
| 41 | R | N | N | N | R | N | R | R | R | N | N | N | N | R | N | N | N | R | N | R | R | N | N | ||
| 53 | N | N | N | N | R | R | R | N | N | N | R | R | N | N | R | R | R | R | N | N | N | R | R | ||
| 61 | R | N | N | N | N | R | N | N | N | R | N | N | R | R | R | N | N | N | R | N | R | N | R | ||
| 73 | N | R | R | R | N | N | N | N | R | R | N | N | R | N | N | N | N | R | R | N | R | R | R | ||
| 89 | N | R | R | R | N | R | N | N | N | N | R | N | N | R | N | R | N | N | N | R | N | R | R | ||
| 97 | N | R | N | N | N | R | R | R | N | N | R | N | R | N | N | N | N | N | N | R | R | R | R | ||
new new new table
edit| q | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 83 | 79 | 71 | 67 | 59 | 47 | 43 | 31 | 23 | 19 | 11 | 7 | 3 | 5 | 13 | 17 | 29 | 37 | 41 | 53 | 61 | 73 | 89 | 97 | ||
| p | 83 | N | N | N | R | N | N | R | R | N | R | R | R | N | N | R | R | R | R | N | R | N | N | N | |
| 79 | R | N | R | N | N | N | R | R | R | R | N | N | R | R | N | N | N | N | N | N | R | R | R | ||
| 71 | R | R | N | N | N | R | N | N | R | N | N | R | R | N | N | R | R | N | N | N | R | R | N | ||
| 67 | R | N | R | R | R | N | N | R | R | N | N | N | N | N | R | R | R | N | N | N | R | R | N | ||
| 59 | N | R | R | N | N | N | N | N | R | N | R | R | R | N | R | R | N | R | R | N | N | N | N | ||
| 47 | R | R | R | N | R | N | N | N | N | N | R | R | N | N | R | N | R | N | R | R | N | R | R | ||
| 43 | R | R | N | R | R | R | R | R | N | R | N | N | N | R | R | N | N | R | R | N | N | N | R | ||
| 31 | N | N | R | R | R | R | N | N | R | N | R | N | R | N | N | N | N | R | N | N | N | N | R | ||
| 23 | N | N | R | N | R | R | N | R | N | N | N | R | N | R | N | R | N | R | N | N | R | N | N | ||
| 19 | R | N | N | N | N | R | R | N | R | R | R | N | R | N | R | N | N | N | N | R | R | N | N | ||
| 11 | N | N | R | R | R | R | N | R | R | N | N | R | R | N | N | N | R | N | R | N | N | R | R | ||
| 7 | N | R | R | R | N | N | R | N | R | N | R | N | N | N | N | R | R | N | R | N | N | N | N | ||
| 3 | N | R | N | R | N | N | R | R | N | R | N | R | N | R | N | N | R | N | N | R | R | N | R | ||
| 5 | N | R | R | N | R | N | N | R | N | R | R | N | N | N | N | R | N | R | N | R | N | R | N | ||
| 13 | N | R | N | N | N | N | R | N | R | N | N | N | R | N | R | R | N | N | R | R | N | N | N | ||
| 17 | R | N | N | R | R | R | R | N | N | R | N | N | N | N | R | N | N | N | R | N | N | R | N | ||
| 29 | R | N | R | R | R | N | N | N | R | N | N | R | N | R | R | N | N | N | R | N | N | N | N | ||
| 37 | R | N | R | R | N | R | N | N | N | N | R | R | R | N | N | N | N | R | R | N | R | N | N | ||
| 41 | R | N | N | N | R | N | R | R | R | N | N | N | N | R | N | N | N | R | N | R | R | N | N | ||
| 53 | N | N | N | N | R | R | R | N | N | N | R | R | N | N | R | R | R | R | N | N | N | R | R | ||
| 61 | R | N | N | N | N | R | N | N | N | R | N | N | R | R | R | N | N | N | R | N | R | N | R | ||
| 73 | N | R | R | R | N | N | N | N | R | R | N | N | R | N | N | N | N | R | R | N | R | R | R | ||
| 89 | N | R | R | R | N | R | N | N | N | N | R | N | N | R | N | R | N | N | N | R | N | R | R | ||
| 97 | N | R | N | N | N | R | R | R | N | N | R | N | R | N | N | N | N | N | N | R | R | R | R | ||