programmingpraxis
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Python,
pasted
on Feb 1:
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# prime.py -- modest prime number library
def primes(n):
"""
list of primes not exceeding n in ascending
order; assumes n is an integer greater than
1; uses Sieve of Eratosthenes
"""
m = (n-1) // 2
b = [True] * m
i, p, ps = 0, 3, [2]
while p*p < n:
if b[i]:
ps.append(p)
j = 2*i*i + 6*i + 3
while j < m:
b[j] = False
j = j + 2*i + 3
i += 1; p += 2
while i < m:
if b[i]:
ps.append(p)
i += 1; p += 2
return ps
def is_prime(n):
"""
False if n is provably composite, else
True if n is probably prime; assumes n
is an integer greater than 1; uses
Miller-Rabin test on prime bases < 100
"""
ps = [2,3,5,7,11,13,17,19,23,29,31,37,41,
43,47,53,59,61,67,71,73,79,83,89,97]
def is_spsp(n, a):
d, s = n-1, 0
while d%2 == 0:
d /= 2; s += 1
if pow(a,d,n) == 1:
return True
for r in xrange(s):
if pow(a, d*pow(2,r), n) == n-1:
return True
return False
if n in ps: return True
for p in ps:
if not is_spsp(n,p):
return False
return True
def factors(n):
"""
list of prime factors of n in ascending
order; assumes n is an integer, may be
positive, zero or negative; uses Pollard's
rho algorithm with Floyd's cycle finder
"""
def gcd(a,b):
while b: a, b = b, a%b
return abs(a)
def facts(n,c,fs):
f = lambda(x): (x*x+c) % n
if is_prime(n): return fs+[n]
t, h, d = 2, 2, 1
while d == 1:
t = f(t); h = f(f(h))
d = gcd(t-h, n)
if d == n:
return facts(n, c+1, fs)
if is_prime(d):
return facts(n//d, c+1, fs+[d])
return facts(n, c+1, fs)
if -1 <= n <= 1: return [n]
if n < -1: return factors(-n)+[-1]
fs = []
while n%2 == 0:
n = n//2; fs = fs+[2]
if n == 1: return fs
return sorted(facts(n,1,fs))
# simple demonstrations
print primes(100)
print is_prime(pow(2,89)-1)
print factors(600851475143)
# divisors
def numdiv(n):
fs = factors(n)
f = fs.pop(0); d = 1; x = 2
while fs:
if f == fs[0]:
x += 1
else:
d *= x; x = 2
f = fs.pop(0)
return d * x
print numdiv(76576500)
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Output:
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[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
True
[71L, 839L, 1471L, 6857L]
576
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