Legendre symbol

### 7.1.25 Legendre symbol

If n is prime, the Legendre symbol of a∈ℤ is written (a/n) and defined by:

| ⎛ ⎜ ⎜ ⎝ |
a
n
| ⎞ ⎟ ⎟ ⎠ | = | ⎧ ⎪ ⎨ ⎪ ⎩ |
0 | if a≡ 0(mod n )
1 | if a ≢0 (mod n ) and a≡ b2 (mod n )
−1 | if a ≢0 (mod n ) and a ≢b2 (mod n )

The Legendre symbol satisfies the following properties.

- If n is prime, then an−1/2≡ (a/n) (mod n ) .
- The following holds for odd and positive numbers p and k:

  | ⎛ ⎜ ⎜ ⎝ |
  p
  q
  | ⎞ ⎟ ⎟ ⎠ | . | ⎛ ⎜ ⎜ ⎝ |
  q
  p
  | ⎞ ⎟ ⎟ ⎠ | =(−1) |
  p−1
  2
  | .(−1) |
  q−1
  2
  | , | ⎛ ⎜ ⎜ ⎝ |
  2
  p
  | ⎞ ⎟ ⎟ ⎠ | =(−1) |
  p2−1
  8
  | , | ⎛ ⎜ ⎜ ⎝ |
  −1
  p
  | ⎞ ⎟ ⎟ ⎠ | =(−1) |
  p−1
  2
  | .

The legendre_symbol command computes the Legendre symbol.

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

#### Examples

legendre_symbol(26,17)

1 |  |  |  |  |  |  |  |  |  |

legendre_symbol(27,17)

−1 |  |  |  |  |  |  |  |  |  |

legendre_symbol(34,17)

0 |  |  |  |  |  |  |  |  |  |