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The Fundamental Theorem of Arithmetic says the following:

For every integer x note 1, there exists a unique, finite series of prime numbers p = {p1, p2,...pn}, such that Π p = x note 2.
For a prime number, this list p is simply that prime, and none others.

This theorem, though it seems trivial, is important to several aspects of math. Once you list the prime factors of a number, you can list all its divisors. Finding the GCD of two numbers is easy: take the intersection of their prime factorizations. This theorem is also important to proving the infiniteness of primes, among other things.

This theorem was first noticed by Euclid, but was first proven by Gauss in his Disquisitiones Arithmeticae. A minimal contradictory proof is as follows.

Assume that there exist numbers that cannot be expressed as the product of a series of primes. Call the smallest of them n. n must be greater than one.again, note 1 Since a prime number is the product of itself (and one), n cannot be prime. The opposite of prime is composite, which means that n=ab where a and b are both positive integers smaller than n. Since n is the smallest number that cannot be written as a product of primes, a and b can. But, since n=ab, n, is also the product of a and b's prime series, which is itself a prime series (just put the two back to back). This contradicts the fact that n cannot be the product of primes.


Note 1: The number one is a special case when it comes to prime numbers and multiplication. Since one is not the product of anything (other than one), it is neither prime nor composite. Kinda like how zero is neither positive nor negative. One is included in this theorem normally, by way of its prime factorization being an empty set.

Note 2: Π is the much the same as Σ. It is the product of all the numbers in the series. Π a,b,c = abc

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