The Fundamental Theorem of Arithmetic

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The fundamental theorem of arithmetic is also called the prime factorisation theorem and the unique factorisation theorem.

It states that for any positive integer greater than or equal to 2, that the number is either prime itself or it can be written as the product of prime numbers.

This product of primes is known as the prime factorisation of the number.

The theorem goes further and says that not only does the prime factorisation exist but that it is also unique up to rearrangement of the order that the primes appear in the product.

In this video we prove this. The proof relies on Euclid’s Lemma. So please be familiar with this before watching.
  1. Elliot Nicholson
  2. The fundamental theorem of arithmetic
  3. fundamental theorem of arithmetic
  4. The Fundamental Theorem of Arithmetic
  5. the fundamental theorem of arithmetic
  6. fundamental theorem of arithmetic proof
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Amazingly clear explanation! Absolutely perfect.

vethya
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I know the theorem holds, but what I don't buy into this types of proofs is right in the beginning: either x is prime or it can be factorized into primes. Well, my gripe with it is that you can't use the fact that a number has a prime factorization to prove a number has a prime factorization. By that I don't mean a problem with the method of contradiction, that's fine, I mean using the result to prove the result.
The premise should be: either x is prime, or it is not. Then you work in the 'not' case, how and why it is a product of primes instead of something else.

kabzebrowski
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