# programming with prime numbers

def primes(n):
    if type(n) != int and type(n) != long:
        raise TypeError('must be integer')
    if n < 2:
        raise ValueError('must be greater than one')
    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 td_prime(n, limit=1000000):
    if type(n) != int and type(n) != long:
        raise TypeError('must be integer')
    if n % 2 == 0:
        return n == 2
    d = 3
    while d * d <= n:
        if limit < d:
            raise OverflowError('limit exceeded')
        if n % d == 0:
            return False
        d += 2
    return True

def td_factors(n, limit=1000000):
    if type(n) != int and type(n) != long:
        raise TypeError('must be integer')
    fs = []
    while n % 2 == 0:
        fs += [2]
        n /= 2
    if n == 1:
        return fs
    f = 3
    while f * f <= n:
        if limit < f:
            raise OverflowError('limit exceeded')
        if n % f == 0:
            fs += [f]
            n /= f
        else:
            f += 2
    return fs + [n]

def is_prime(n):
    if type(n) != int and type(n) != long:
        raise TypeError('must be integer')
    if n < 2:
        return False
    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
        t = pow(a,d,n)
        if t == 1:
            return True
        while s > 0:
            if t == n-1:
                return True
            t = (t*t) % n
            s -= 1
        return False
    if n in ps: return True
    for p in ps:
        if not is_spsp(n,p):
            return False
    return True

def rho_factors(n):
    if type(n) != int and type(n) != long:
        raise TypeError('must be integer')
    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 [-1] + rho_factors(-n)
    fs = []
    while n%2 == 0:
        n = n//2; fs = fs+[2]
    if n == 1: return fs
    return sorted(facts(n,1,fs))
    
print primes(100)
print len(primes(1000000))
print td_prime(600851475143)
print td_factors(600851475143)
print is_prime(600851475143)
print is_prime(2305843009213693951)
print rho_factors(600851475143)

[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]
78498
False
[71, 839, 1471, 6857L]
False
True
[71L, 839L, 1471L, 6857L]