Jacobi symbol

### 7.1.26 Jacobi symbol

The Jacobi symbol is a generalization of the Legendre symbol (a/n) when n is not prime. Let

n=p1α 1⋯ pkα k

be the prime factorization of n. The Jacobi symbol of a is defined by:

| ⎛ ⎜ ⎜ ⎝ |
a
n
| ⎞ ⎟ ⎟ ⎠ | = | ⎛ ⎜ ⎜ ⎝ |
a
p1
| ⎞ ⎟ ⎟ ⎠ |
α1
| … | ⎛ ⎜ ⎜ ⎝ |
a
pk
| ⎞ ⎟ ⎟ ⎠ |
α k

Where the left hand side is the Jacobi symbol and the right hand side contains Legendre symbols. The jacobi_symbol command computes the Jacobi symbol.

- jacobi_symbol takes two arguments: a and n, integers.
- jacobi_symbol(a,n) returns the Jacobi symbol (a/n).

#### Examples

jacobi_symbol(25,12)

1 |  |  |  |  |  |  |  |  |  |

jacobi_symbol(35,12)

−1 |  |  |  |  |  |  |  |  |  |

jacobi_symbol(33,12)

0 |  |  |  |  |  |  |  |  |  |