Prime Factorization

Prime factorization or integer factorization of a number is breaking a number down into the set of prime numbers which multiply together to result in the original number. This is also known as prime decomposition.

Prime Factorization using Trial Division

Following are the steps to find all prime factors.

While n is divisible by 2, print 2 and divide n by 2
After step 1, n must be odd. Now start a loop from i = 3 to square root of n. While i divides n, print i and divide n by i, increment i by 2 and continue
If n is a prime number and is greater than 2, then n will not become 1 by above two steps. So print n if it is greater than 2

Prime Factors Tree

Using a prime factorization tree to see the work, prime decomposition of 100 = 2 x 2 x 5 x 5 looks like this:

Optimization 1: running the loop till `sqrt(n)`

Now the main part is, the loop runs till square root of n not till. To prove that this optimization works, let us consider the following property of composite numbers.

Every composite number has at least one prime factor less than or equal to square root of itself.

This property can be proved using counter statement. Let a and b be two factors of n such that a*b = n. If both are greater than √n, then a.b > √n,* √n, which contradicts the expression a * b = n.

Running Time : sqrt(n)

Code

def primeFactors(n):

    # Print the number of two's that divide n
    while n % 2 == 0:
        print 2,
        n = n / 2

    # n must be odd at this point
    # so a skip of 2 ( i = i + 2) can be used
    for i in range(3,int(math.sqrt(n))+1,2):

        # while i divides n , print i ad divide n
        while n % i== 0:
            print i,
            n = n / i

    # Condition if n is a prime
    # number greater than 2
    if n > 2:
    print n

For multiple queries we can use Sieve of Eratosthenes for giving result in O(log n)

Sieve of Eratosthenes, (for primality test)

def sieve(n):
    prime = [True for i in range(n + 1)]
    p = 2
    while (p * p <= n):

        # If prime[p] is not changed, then it is a prime
        if (prime[p] == True):

            # Update all multiples of p
            for i in range(p * 2, n + 1, p):
                prime[i] = False
        p += 1
    prime[0]= False
    prime[1]= False

    lst = []
    for p in range(n + 1):
        if prime[p]:
            lst.append(p)

return lst

Sieve of Atkin

Factors of a number

# find all factors of a given number
import math

def printDivisors(n):

    # Note that this loop runs till square root
    i = 1
    fac = []
    while i <= math.sqrt(n):

        if n % i == 0:

            # If divisors are equal, print only one
            if (n // i == i):
                fac.append(i)
            else:
                # Otherwise print both
                fac.append([i, n//i])

        i = i + 1

    return fac

assert printDivisors(100) == [[1, 100], [2, 50], [4, 25], [5, 20], 10]

Time Complexity : O(sqrt(n))

Auxiliary Space : O(1)

https://www.geeksforgeeks.org/find-divisors-natural-number-set-1

Others

https://en.wikipedia.org/wiki/Shor%27s_algorithm

Prime Factorization