Python：分解质因数

把一个合数用质因数相乘的形式表示出来，叫做分解质因数. 分解质因数常见方法是短除法，也可以用Python实现.

给出三种分解质因数的代码：

Z=int(input('x{x∈Z}='))
print(Z,'= ',end='')

if Z<0:
    Z=abs(Z)
    print('-',end='')

flag=0
if Z<=1:
    print(Z)
    flag=1


while True:
    if flag:
        break
    for i in range(2, int(Z+1)):
        if Z%i==0:
            print("%d"%i, end='')
            if Z==i:
                flag=1
                break
            print('×', end='')
            Z/=i
            break

n = input("合数：")
if n.isdigit():
    n = int(n)
else:
    print("输入非法，请输入一个合数")
    exit()
 
if n < 2:
    print("请输入一个大于2的合数")
    exit()
 
def isZhishu(n):  # 判断是否是质数
    for i in range(2, n):
        if n % i == 0:  # 不是质数
            return False
    else:
        return True
 
l0 = []
def fenjie(n):
    i = 2
    while i < n + 1:
        if n % i == 0:
            l0.append(i)
            n /= i
        else:
            i += 1
 
 
if not isZhishu(n):
    fenjie(n)
    str0 = ''
    for i in l0:
        str0 = str0 + str(i) + "*"
    str0 = str0[:-1]  # 去掉最后一个星号
    print("%s=%s" % (n, str0))
else:
    print("%s是一个质数，请输入一个合数" %n)

#MillerRabin素数判定,结合Pollard_rho递归分解,效率极高
 
import random
from collections import Counter
 
def gcd(a, b):
    if a == 0:
        return b
    if a < 0:
        return gcd(-a, b)
    while b > 0:
        c = a % b
        a, b = b, c
    return a
        
def mod_mul(a, b, n):
    result = 0
    while b > 0:
        if (b & 1) > 0:
            result = (result + a) % n
        a = (a + a) % n
        b = (b >> 1)
    return result
     
def mod_exp(a, b, n):
    result = 1
    while b > 0:
        if (b & 1) > 0:
            result = mod_mul(result, a, n)
        a = mod_mul(a, a, n)
        b = (b >> 1)
    return result
     
def MillerRabinPrimeCheck(n):
    if n in {2, 3, 5, 7, 11}:
        return True
    elif (n == 1 or n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0 or n % 11 == 0):
        return False
    k, u = 0, n - 1
    while not (u & 1) > 0:
        k += 1
        u = (u >> 1)
    random.seed(0)
    s = 5
    for i in range(s):
        x = random.randint(2, n - 1)
        if x % n == 0:
            continue
        x = mod_exp(x, u, n)
        pre = x
        for j in range(k):
            x = mod_mul(x, x, n)
            if (x == 1 and pre != 1 and pre != n - 1):
                return False
            pre = x
        if x != 1:
            return False
        return True
         
def Pollard_rho(x, c):
    (i, k) = (1, 2)
    x0 = random.randint(0, x)
    y = x0
    while 1:
        i += 1
        x0 = (mod_mul(x0, x0, x) + c) % x
        d = gcd(y - x0, x)
        if d != 1 and d != x:
            return d
        if y == x0:
            return x
        if i == k:
            y = x0
            k += k
 
def PrimeFactorsListGenerator(n):
    result = []
    if n <= 1:
        return None
    if MillerRabinPrimeCheck(n):
        return [n]
    p = n
    while p >= n:
        p = Pollard_rho(p, random.randint(1, n - 1))
    result.extend(PrimeFactorsListGenerator(p))
    result.extend(PrimeFactorsListGenerator(n // p))
    return result
 
def PrimeFactorsListCleaner(n):
    return Counter(PrimeFactorsListGenerator(n))
                   
PrimeFactorsListCleaner(1254000000)