def jacobi_symbol(m, n):
"""
Returns the product of the legendre_symbol(m, p)
for all the prime factors p of n.
Returns
=======
1. 0 if m cong 0 mod(n)
2. 1 if x**2 cong m mod(n) has a solution
3. -1 otherwise
Examples
========
>>> from sympy.ntheory import jacobi_symbol, legendre_symbol
>>> from sympy import Mul, S
>>> jacobi_symbol(45, 77)
-1
>>> jacobi_symbol(60, 121)
1
The relationship between the jacobi_symbol and legendre_symbol can
be demonstrated as follows:
>>> L = legendre_symbol
>>> S(45).factors()
{3: 2, 5: 1}
>>> jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1
True
"""
m, n = int_tested(m, n)
if not n % 2:
raise ValueError("n should be an odd integer")
if m < 0 or m > n:
m = m % n
if not m:
return int(n == 1)
if n == 1 or m == 1:
return 1
if igcd(m, n) != 1:
return 0
j = 1
s = trailing(m)
m = m >> s
if s % 2 and n % 8 in [3, 5]:
j *= -1
while m != 1:
if m % 4 == 3 and n % 4 == 3:
j *= -1
m, n = n % m, m
s = trailing(m)
m = m >> s
if s % 2 and n % 8 in [3, 5]:
j *= -1
return j
def jacobi_symbol(m, n):
"""
Returns the product of the legendre_symbol(m, p)
for all the prime factors p of n.
Returns
=======
1. 0 if m cong 0 mod(n)
2. 1 if x**2 cong m mod(n) has a solution
3. -1 otherwise
Examples
========
>>> from sympy.ntheory import jacobi_symbol, legendre_symbol
>>> from sympy import Mul, S
>>> jacobi_symbol(45, 77)
-1
>>> jacobi_symbol(60, 121)
1
The relationship between the jacobi_symbol and legendre_symbol can
be demonstrated as follows:
>>> L = legendre_symbol
>>> S(45).factors()
{3: 2, 5: 1}
>>> jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1
True
"""
m, n = int_tested(m, n)
if not n % 2:
raise ValueError("n should be an odd integer")
if m < 0 or m > n:
m = m % n
if not m:
return int(n == 1)
if n == 1 or m == 1:
return 1
if igcd(m, n) != 1:
return 0
j = 1
s = trailing(m)
m = m >> s
if s % 2 and n % 8 in [3, 5]:
j *= -1
while m != 1:
if m % 4 == 3 and n % 4 == 3:
j *= -1
m, n = n % m, m
s = trailing(m)
m = m >> s
if s % 2 and n % 8 in [3, 5]:
j *= -1
return j
def jacobi_symbol(m, n):
"""
Returns 0 if m cong 0 mod(n),
Return 1 if x**2 cong m mod(n) has a solution else return -1.
jacobi_symbol(m, n) is product of the legendre_symbol(m, p) for all the prime factors p of n.
**Examples**
>>> from sympy.ntheory import jacobi_symbol
>>> jacobi_symbol(45, 77)
-1
>>> jacobi_symbol(60, 121)
1
"""
if not n % 2:
raise ValueError("n should be an odd integer")
if m < 0 or m > n:
m = m % n
if igcd(m, n) != 1:
return 0
j = 1
s = trailing(m)
m = m >> s
if s % 2 and n % 8 in [3, 5]:
j = (-1)**s
while m != 1:
if m % 4 == 3 and n % 4 == 3:
j *= -1
m, n = n % m, m
s = trailing(m)
m = m >> s
if s % 2 and n % 8 in [3, 5]:
j = (-1)**s
return j
