The Kronecker symbol is an extension of the Jacobi symbol 
to all integers.
It is variously written as 
or 
(Cohn 1980; Weiss 1998, p. 236) or 
(Dickson 2005). The Kronecker symbol is multiplicative
in both arguments when the denominators are positive. For integers 
and 
and positive integers 
and 
,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
Negative denominators are handled using
 |
(4)
|
and a zero denominator is handled using
 |
(5)
|
The definition for 
is variously written as
 |
(6)
|
or
 |
(7)
|
(Cohn 1980). Cohn's form "undefines" 
for singly even numbers 
and 
. For the binary
quadratic form discriminants 
of quadratic fields, only
is needed.
The Kronecker symbol is implemented in the Wolfram Language as KroneckerSymbol[n, m].
For a fundamental discriminant 
, the Kronecker symbol 
is a primitive real number theoretic character of conductor 
. Every nonprincipal primitive real number theoretic character arises from
a unique fundamental discriminant. The
principal character modulo 1 is represented by 
(Ayoub 1963, Goldmakher).
The illustration above and table below summarize 
for 
, 2, ... and small 
.
 | OEIS | period |  |
 | A109017 | 24 | 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0,  , 0, 0, 0,  , 0,  , 0, ... |
 | 0 | 1,
 , 1, 1, 0,  , 1,  , 1, 0,  , 1,  ,  , 0, 1,  ,  ,  , 0, ... | |
 | 4 | 1,
0,  ,
0, 1, 0,  ,
0, 1, 0,  ,
0, 1, 0,  ,
0, 1, 0,  ,
0, ... | |
 | 3 | 1,
 , 0, 1,  , 0, 1,  , 0, 1,  , 0, 1,  , 0, 1,  , 0, 1,  , ... | |
 | 8 | 1,
0, 1, 0,  ,
0,  ,
0, 1, 0, 1, 0,  ,
0,  ,
0, 1, 0, 1, 0, ... | |
 | A034947 | 1, 1,  , 1, 1,  ,  , 1, 1, 1,  ,  , 1,  ,  , 1, 1, ... | |
| 0 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... | ||
| 1 | 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... | |
| 2 | A091337 | 8 | 1,
0,  ,
0,  ,
0, 1, 0, 1, 0,  ,
0,  ,
0, 1, ... |
| 3 | A091338 | 1,  , 0, 1,  , 0,  ,  , 0, 1, 1, 0, 1, 1, 0, ... | |
| 4 | A000035 | 2 | 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ... |
| 5 | A080891 | 5 | 1,  ,  , 1, 0, 1,  ,  , 1, 0, 1,  ,  , 1, 0, ... |
| 6 | 24 | 1, 0, 0, 0, 1, 0,  , 0, 0, 0,  , 0,  , 0, 0, 0,  , 0, 1, 0, ... |
For a fundamental discriminant 
, the period of 
equals 
. For 
, 
, ..., the periods of 
are 0, 8, 3, 4, 0, 24, 7, 8, 0, 40, 11, 6, ... (OEIS A117889) and for 
, 2, ... they are 1, 8, 0, 2, 5, 24, 0, 8, 3, 40, 0, 12,
... (OEIS A117888). Here, 0 indicates that
the sequence is not periodic.
