What is Fundamental theorem of arithmetic?
The '''Fundamental theorem of arithmetic''' (also called the '''unique factorization theorem''') is a [[theorem]] of [[number theory]]. The [[theorem]] says that ''every'' positive [[integer]] greater than 1 can be written as a [[product (mathematics)|product]] of [[prime number]]s (or the integer is itself a prime number). The theorem also says that there is only one way to write the number. If two people found two different ways to write the number, the only thing that can be different is the order in which the primes are written. For example, we can write: :6936 = 2<sup>3</sup> · 3 · 17<sup>2</sup> &nbsp; or &nbsp; 1200 = 2<sup>4</sup> · 3 · 5<sup>2</sup> and if somebody else finds ''another'' way to write 6936 or 1200 as product of prime numbers, we can put those prime numbers in the right order and find out that it is the same as what we have here. Finding the prime numbers is called [[factorization]]. This theorem can be used in [[cryptography]]. == Proof == The first person who proved the [[theorem]] was [[Euclid]]. The first detailed and correct [[wikt:proof|proof]] was in the ''[[Disquisitiones Arithmeticae]]'' by [[Carl Friedrich Gauss]]. Some people may think that the theorem is true everywhere. However, the theorem is ''not'' true in more general number systems, like [[:en:Algebraic_integer|algebraic integers]]. This was first mentioned by [[Ernst Kummer]] in 1843, in his work on [[Fermat's last theorem]]. For more information about that: read [[:en:Algebraic_number_theory|algebraic number theory]]. The proof consists of two parts: ''first'' we show that every number can be written as a product of primes; ''second'' we show that if we write a number as a product of primes for a second time, then the two lists of prime numbers must be the same. === First part of the proof === We show that if ''not'' every number greater than 1 can be written as a product of primes, we end up in some kind of impossibility. So after that we conclude that it ''must'' be true that every number can be written as