If you want to show there are infinitely many primes, one way is to first note that every integer greater than 1 has a prime factor. This is because if an integer n is prime, n is a prime factor of itself, and if n is not prime then it must have a smaller factor m other than 1, 1< m < n. If m is also not prime, it too must have a smaller factor other than 1, and you can keep playing this game but there are only so many integers between 1 and n so eventually you’ll get to a factor of n that has no smaller factors of its own other than 1, which means it is prime.
Let’s now suppose there is only a finite number of primes, we’ll try to show that this assumption leads to nonsense so can’t be possible.
We can multiply any finite number of integers together to get a new integer. Let’s multiply all of the primes together to get a new number M. Then M + 1 gives a remainder of 1 when you divide by any prime number. Since dividing by a factor will always give a remainder of 0, none of the prime numbers can be a factor of M + 1. So M + 1 is an imteger bigger than 1 with no prime factors. This is impossible, so there must be a mistake somewhere in this argument.
The only thing we said that we’re not 100% sure is true was that there are a finite number of primes, so that has to be our mistake. So there must be infinitely many prime numbers.
And your irrefutable proof is…?
If you want to show there are infinitely many primes, one way is to first note that every integer greater than 1 has a prime factor. This is because if an integer n is prime, n is a prime factor of itself, and if n is not prime then it must have a smaller factor m other than 1, 1< m < n. If m is also not prime, it too must have a smaller factor other than 1, and you can keep playing this game but there are only so many integers between 1 and n so eventually you’ll get to a factor of n that has no smaller factors of its own other than 1, which means it is prime.
Let’s now suppose there is only a finite number of primes, we’ll try to show that this assumption leads to nonsense so can’t be possible.
We can multiply any finite number of integers together to get a new integer. Let’s multiply all of the primes together to get a new number M. Then M + 1 gives a remainder of 1 when you divide by any prime number. Since dividing by a factor will always give a remainder of 0, none of the prime numbers can be a factor of M + 1. So M + 1 is an imteger bigger than 1 with no prime factors. This is impossible, so there must be a mistake somewhere in this argument.
The only thing we said that we’re not 100% sure is true was that there are a finite number of primes, so that has to be our mistake. So there must be infinitely many prime numbers.