# Jacobi Symbol (Erlang)

Other implementations: Erlang | Haskell | Python

The Jacobi symbol satisfies the six properties listed below, and these can be used to compute the Jacobi symbol in polynomial time.

1. ```<<property 1>>=
jacobi(A, N) when A rem 2 =:= 0 ->
jacobi(2, N) * jacobi(A div 2 + A rem 2, N);
```
2. For ,.
```<<property 2>>=
jacobi(A, N) when A >= N ->
jacobi(A rem N, N);
```
3. For odd coprimes a and n, .
```<<property 3>>=
jacobi(A, N) when A rem 4 =:= 3, N rem 4 =:= 3 ->
-jacobi(N, A);
jacobi(A, N) ->
jacobi(N, A).
```
4. .
```<<property 4>>=
jacobi(1, _N) ->
1;
```
5. .
```<<property 5>>=
jacobi(2, N) ->
case (N rem 8) of
1 -> 1;
3 -> -1;
5 -> -1;
7 -> 1
end;
```
6. .
```<<property 6>>=
jacobi(0, _N) ->
0;
```

From these properties we can come up with a polynomial time algorithm.

The function always complete as property 1 can be used to remove all even factors, and these will end at property 4. The odd factors will be reduced by property 2 if and if a < n property 3 will swap a and n and property 2 will again reduce a until it matches property 4 or property 6.

```<<jacobi symbol>>=
property 4
property 5
property 6
property 1
property 2
property 3
```

This is an escript, it can be run from the command line on unix type computers.

```<<jacobi>>=
#!/usr/bin/env escript
main([A, P]) ->
io:fwrite("~p~n", [xjacobi(string:to_integer(A), string:to_integer(P))]).
xjacobi({A, _}, {B, _}) ->
jacobi(A, B).
jacobi symbol
```