For every integer N >= 1, there is a unique bag of prime numbers whose product is N.
All the elements in a bag must be prime. If N is itself prime, the bag for N is just {N}.
The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.
Because both the system’s privacy and the security of digital money depend on encryption, a breakthrough in mathematics or computer science that defeats the cryptographic system could be a disaster. The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers.
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# Python
def factor(N):
for d in range(2, N-1):
if N % d == 0: return d
As:
it suffices to search form 2 to <math>\sqrt{N}</math>
# Python
def factor(N):
for d in range(2, intsqrt(N):
if N % d == 0: return d
Find integers a and b such that: <MATH>a^2 − b^2 = N</MATH> Then: <MATH>(a − b)(a + b) = N</MATH> so a − b and a + b are divisors ideally non-trivial
Naive approach to find such integers:
For large N, it takes too long to find a good integer a