def aks(n):
"""
AKS ( Cyclotomic Congruence Test ) primality test for a natural number.
Input: a natural number n ( n > 1 ).
Output: n is prime => return True
n is not prime => return False.
"""
import nzmath.multiplicative as multiplicative
if arith1.powerDetection(n)[1] != 1: #Power Detection
return False
lg = math.log(n, 2)
k = int(lg * lg)
start = 3
while 1:
d = gcd.gcd(start, n)
if 1 < d < n:
return False
x = n % start
N = x
for i in range(1, k + 1):
if N == 1:
break
N = (N * x) % start
if i == k:
r = start
break
start += 1
d = gcd.gcd(r, n)
if 1 < d < n:
return False
if n <= r:
return True
e = multiplicative.euler(r) #Cyclotomic Conguence
e = math.sqrt(e)
e = int(e * lg)
for b in range(1, e + 1):
f = array_poly.ArrayPolyMod([b, 1], n)
total = array_poly.ArrayPolyMod([1], n)
count = n
while count > 0:
if count & 1:
total = total * f
total = _aks_mod(total, r)
f = f.power()
f = _aks_mod(f, r)
count = count >> 1
total_poly = total.coefficients_to_dict()
if total_poly != {0: b, n % r: 1}:
return False
return True
def log(x, base=None, err=defaultError):
"""
log(x) returns logarithm of a positive number x. If an additional
argument base is given, it returns logarithm of x to the base.
"""
if isinstance(x, complex):
raise TypeError("real.log is not for complex numbers.")
if x <= 0:
raise ValueError("log is not defined for %s" % str(x))
if err <= defaultError:
if base:
d = log(base, err=err)
else:
d = 1
rx = rational.Rational(x)
upper = rational.Rational(4, 3)
lower = rational.Rational(2, 3)
shift = 0
while rx > upper:
rx /= 2
shift += 1
while rx < lower:
rx *= 2
shift -= 1
if rx == 1:
return shift * _log2(err) / d
value = oldvalue = 0
for term in log1piter(rx - 1):
value += term
if err.nearlyEqual(value, oldvalue):
break
oldvalue = +value
if shift != 0:
return (value + shift * _log2(err)) / d
else:
if base:
value = rational.Rational(math.log(x, base))
else:
value = rational.Rational(math.log(x))
return value
