eli5 How does Euclid’s theorem prove that there are infinite prime numbers.

Question

eli5 How does Euclid’s theorem prove that there are infinite prime numbers.

In: Mathematics

Answer 1

if there are a finite number of primes, you can put all of them in a list.

if you take this list, and multiply it together you get a huge highly composit number. If you add 1 to it, it is now an even bigger number that isnt divisible by anything in the list of primes. This is the definition of a prime number, so your finite list was incomplelete.

since this is a contradiction for any finite number of primes, there must not be a finite number of primes, i.e, there must be an infinite number of primes