Il simbolo di Jacobi è una funzione, indicata con 
, definita per m e n interi positivi come 
, dove 
è il simbolo di Legendre e 
è la scomposizione di m in fattori primi. Per convenzione 
.
La definizione è talvolta estesa, definendo 
se m e n hanno un divisore comune maggiore di 1.
E’ la generalizzazione del simbolo di Legendre, dal quale eredita la notazione, al caso di m composto; coincide con esso se m è primo, tuttavia mentre se 
, n non è residuo quadratico modulo m, se 
non è detto che n sia un residuo quadratico modulo m: questo accade solo se n è residuo quadratico modulo ogni primo che divida m.
Dalla definizione segue che:
-
la funzione assume solo i valori –1, 0 e 1;
-
; -
; -
, se a ≡ b mod m.
E’ completamente moltiplicativo, sia rispetto al primo argomento, che rispetto al secondo: 
, 
.
Alcune proprietà, valide per m dispari:
;-
, se m e n sono primi tra loro.
Permette di estendere il teorema di reciprocità quadratica a qualsiasi coppia di numeri naturali dispari m e n primi tra loro: 
, ovvero 
, se m e n sono entrambi della forma 4k + 3, 
altrimenti.
Queste proprietà permettono di calcolarlo in modo efficiente con un algoritmo analogo a quello di Euclide per il calcolo del massimo comun divisore.
Dato che può essere calcolato senza conoscere la scomposizione degli argomenti, fornisce un criterio efficiente per determinare se un numero n dispari è composto: si prende un intero a a caso, primo rispetto a n, e si calcolano 
e 
; se i due valori differiscono, n è composto, se sono uguali n è probabilmente primo (e si può ripetere l’esame con un differente valore di a) (v. pseudoprimi di Eulero – Jacobi).
La tabella seguente mostra i primi valori di 
.
|
m \ n |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
2 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
|
3 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
|
4 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
|
5 |
1 |
–1 |
–1 |
1 |
0 |
1 |
–1 |
–1 |
1 |
0 |
1 |
–1 |
–1 |
1 |
0 |
1 |
–1 |
–1 |
1 |
0 |
|
6 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
7 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
0 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
0 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
|
8 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
|
9 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
|
10 |
1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
11 |
1 |
–1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
0 |
1 |
–1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
|
12 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
|
13 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
–1 |
–1 |
1 |
1 |
–1 |
1 |
0 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
–1 |
|
14 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
|
15 |
1 |
1 |
0 |
1 |
0 |
0 |
–1 |
1 |
0 |
0 |
–1 |
0 |
–1 |
–1 |
0 |
1 |
1 |
0 |
1 |
0 |
|
16 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
|
17 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
–1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
1 |
1 |
0 |
1 |
1 |
–1 |
|
18 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
|
19 |
1 |
–1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
1 |
–1 |
–1 |
–1 |
–1 |
1 |
1 |
–1 |
0 |
1 |
|
20 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
|
21 |
1 |
–1 |
0 |
1 |
1 |
0 |
0 |
–1 |
0 |
–1 |
–1 |
0 |
–1 |
0 |
0 |
1 |
1 |
0 |
–1 |
1 |
|
22 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
|
23 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
–1 |
|
24 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
25 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
|
26 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
|
27 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
|
28 |
1 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
|
29 |
1 |
–1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
–1 |
1 |
–1 |
–1 |
–1 |
1 |
|
30 |
1 |
0 |
0 |
0 |
0 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
|
31 |
1 |
1 |
–1 |
1 |
1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
1 |
1 |
1 |
|
32 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
|
33 |
1 |
1 |
0 |
1 |
–1 |
0 |
–1 |
1 |
0 |
–1 |
0 |
0 |
–1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
|
34 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
|
35 |
1 |
–1 |
1 |
1 |
0 |
–1 |
0 |
–1 |
1 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
1 |
–1 |
–1 |
0 |
|
36 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
37 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
–1 |
–1 |
–1 |
|
38 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
|
39 |
1 |
1 |
0 |
1 |
1 |
0 |
–1 |
1 |
0 |
1 |
1 |
0 |
0 |
–1 |
0 |
1 |
–1 |
0 |
–1 |
1 |
|
40 |
1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
41 |
1 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
1 |
|
42 |
1 |
0 |
0 |
0 |
–1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
43 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
–1 |
1 |
1 |
1 |
–1 |
1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
|
44 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
|
45 |
1 |
–1 |
0 |
1 |
0 |
0 |
–1 |
–1 |
0 |
0 |
1 |
0 |
–1 |
1 |
0 |
1 |
–1 |
0 |
1 |
0 |
|
46 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
|
47 |
1 |
1 |
1 |
1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
1 |
1 |
1 |
–1 |
–1 |
|
48 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
|
49 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
|
50 |
1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
|
51 |
1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
0 |
–1 |
1 |
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
|
52 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
|
53 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
1 |
–1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
|
54 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
55 |
1 |
1 |
–1 |
1 |
0 |
–1 |
1 |
1 |
1 |
0 |
0 |
–1 |
1 |
1 |
0 |
1 |
1 |
1 |
–1 |
0 |
|
56 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
|
57 |
1 |
1 |
0 |
1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
0 |
–1 |
1 |
0 |
1 |
–1 |
0 |
0 |
–1 |
|
58 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
|
59 |
1 |
–1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
1 |
–1 |
–1 |
1 |
–1 |
–1 |
1 |
1 |
1 |
–1 |
1 |
1 |
|
60 |
1 |
0 |
0 |
0 |
0 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
61 |
1 |
–1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
–1 |
1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
1 |
1 |
|
62 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
|
63 |
1 |
1 |
0 |
1 |
–1 |
0 |
0 |
1 |
0 |
–1 |
1 |
0 |
–1 |
0 |
0 |
1 |
–1 |
0 |
–1 |
–1 |
|
64 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
|
65 |
1 |
1 |
–1 |
1 |
0 |
–1 |
1 |
1 |
1 |
0 |
–1 |
–1 |
0 |
1 |
0 |
1 |
–1 |
1 |
–1 |
0 |
|
66 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
67 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
–1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
|
68 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
|
69 |
1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
0 |
–1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
–1 |
1 |
|
70 |
1 |
0 |
–1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
71 |
1 |
1 |
1 |
1 |
1 |
1 |
–1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
1 |
1 |
–1 |
1 |
1 |
1 |
|
72 |
1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
|
73 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
1 |
1 |
–1 |
|
74 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
|
75 |
1 |
–1 |
0 |
1 |
0 |
0 |
1 |
–1 |
0 |
0 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
0 |
|
76 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
|
77 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
0 |
–1 |
1 |
1 |
0 |
–1 |
1 |
0 |
1 |
1 |
1 |
–1 |
1 |
–1 |
|
78 |
1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
0 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
|
79 |
1 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
1 |
1 |
|
80 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
|
81 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
|
82 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
|
83 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
1 |
–1 |
–1 |
–1 |
|
84 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
|
85 |
1 |
–1 |
1 |
1 |
0 |
–1 |
1 |
–1 |
1 |
0 |
–1 |
1 |
–1 |
–1 |
0 |
1 |
0 |
–1 |
1 |
0 |
|
86 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
|
87 |
1 |
1 |
0 |
1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
|
88 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
|
89 |
1 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
1 |
1 |
1 |
1 |
–1 |
–1 |
–1 |
–1 |
1 |
1 |
1 |
–1 |
1 |
|
90 |
1 |
0 |
0 |
0 |
0 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
91 |
1 |
–1 |
–1 |
1 |
1 |
1 |
0 |
–1 |
1 |
–1 |
–1 |
–1 |
0 |
0 |
–1 |
1 |
–1 |
–1 |
1 |
1 |
|
92 |
1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
–1 |
0 |
|
93 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
0 |
1 |
1 |
0 |
1 |
–1 |
|
94 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
|
95 |
1 |
1 |
1 |
1 |
0 |
1 |
–1 |
1 |
1 |
0 |
1 |
1 |
1 |
–1 |
0 |
1 |
–1 |
1 |
0 |
0 |
|
96 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
0 |
0 |
–1 |
0 |
–1 |
0 |
|
97 |
1 |
1 |
1 |
1 |
–1 |
1 |
–1 |
1 |
1 |
–1 |
1 |
1 |
–1 |
–1 |
–1 |
1 |
–1 |
1 |
–1 |
–1 |
|
98 |
1 |
0 |
–1 |
0 |
–1 |
0 |
0 |
0 |
1 |
0 |
–1 |
0 |
–1 |
0 |
1 |
0 |
1 |
0 |
–1 |
0 |
|
99 |
1 |
–1 |
0 |
1 |
1 |
0 |
–1 |
–1 |
0 |
–1 |
0 |
0 |
–1 |
1 |
0 |
1 |
–1 |
0 |
–1 |
1 |
|
100 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |