As I understand it, the proof that there are an infinite number of prime numbers is that if you take the factorial of the largest prime you can think of, say P!, it will be divisible by every number up to P!, as P! is equal to the product of all those numbers.

As all numbers are divisible by prime numbers, if you have P! + 1, that number can not be divisible by the previous set of number so it must be either a prime number or divisible by a higher prime number than P.

I was also told that the larger the numbers get the fewer prime numbers you seem to get or the frequency of prime numbers seems to diminish.

That got me thinking, the frequency of prime numbers can’t continue to decrease otherwise the frequency or distance between two primes would have to reach infinity at some point. Then there would be an infinite amount of composite numbers after the last prime, which obviously can’t be the case if there are an infinite number of primes.

So that would suggest that there must be a limit to the distance between any two prime numbers as it can’t be infinity or one would never get to the next prime.

However it is possible to prove that one can have an infinite sequence of composite numbers. For example 100! + 2 is divisible by 100!, 1 and 2 thus it is a composite. And 100! + 3 is divisible by 100!, 1 and 3 and so on. I could replace the 100 with 1,000,000 and have 1,000,000 sequential composite numbers (or 999,999 I think it would be). Thus I could replace the 100 with an infinite amount of numbers.

Thus one can have an infinite number of sequential composite numbers, which would mean no more primes!

So what don’t I understand! ‘cos that’s doing my head in!

As all numbers are divisible by prime numbers, if you have P! + 1, that number can not be divisible by the previous set of number so it must be either a prime number or divisible by a higher prime number than P.

I was also told that the larger the numbers get the fewer prime numbers you seem to get or the frequency of prime numbers seems to diminish.

That got me thinking, the frequency of prime numbers can’t continue to decrease otherwise the frequency or distance between two primes would have to reach infinity at some point. Then there would be an infinite amount of composite numbers after the last prime, which obviously can’t be the case if there are an infinite number of primes.

So that would suggest that there must be a limit to the distance between any two prime numbers as it can’t be infinity or one would never get to the next prime.

However it is possible to prove that one can have an infinite sequence of composite numbers. For example 100! + 2 is divisible by 100!, 1 and 2 thus it is a composite. And 100! + 3 is divisible by 100!, 1 and 3 and so on. I could replace the 100 with 1,000,000 and have 1,000,000 sequential composite numbers (or 999,999 I think it would be). Thus I could replace the 100 with an infinite amount of numbers.

Thus one can have an infinite number of sequential composite numbers, which would mean no more primes!

So what don’t I understand! ‘cos that’s doing my head in!

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