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Integer Factorization

## Overview

Given a composite integer
$n$
, can it be decomposed as a product of smaller integers (hopefully as a unique product of prime factors)?
As easy as it may sound, integer factorization in polynomial time on a classical computer stands one of the unsolved problems in computation for centuries!
Lets start dumb, all we need to do is check all the numbers
$1 < p < n$
such that
$p|n$
or programmatically n%p==0
def factors(n):
divisors = []
for p in range(1,n):
if n%p==0:
divisors.append(p)
return divisors
Seems like its an
$O(n)$
algorithm! whats all the deal about? By polynomial time, we mean polynomial time in
$b$
when
$n$
is a b-bit number, so what we looking at is actually a
$O(2^b)$
which is actually exponential (which everyone hates)
Now taking a better look at it, one would realize that a factor of
$n$
can't be bigger than
$\sqrt{n}$
Other observation would be, if we already checked a number (say 2) to not be a divisor, we dont need to check any multiple of that number since it would not be a factor.
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