But then the productĬan be written as a product of primes as well. So it must be possible to write a and b as products of primes, because they are both smaller than n. But: n was the smallest number that can not be written as a product of primes. Where both a and b are positive integers that are of course smaller than n. Further, it cannot be a prime number, because a prime number is a 'product' of a single prime: itself. One of these numbers must be the smallest: let's call it n. In that case we ask him/her to mention all the numbers, greater than 1, that can not be written as a product of primes. So, now see what happens when somebody says that he/she knows a positive integer, greater than 1, which can not be written as a product of primes. So after that we conclude that it must be true that every number can be written as a product of primes. 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. 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.įirst part of the proof For more information about that: read algebraic number theory.
This was first mentioned by Ernst Kummer in 1843, in his work on Fermat's last theorem.
However, the theorem is not true in more general number systems, like algebraic integers. Some people may think that the theorem is true everywhere. The first detailed and correct proof was in the Disquisitiones Arithmeticae by Carl Friedrich Gauß. The first person who proved the theorem was Euclid.
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