[UCANSEE2]

As far as proving an infinity, well good luck with that.

I would suggest that if you stipulate that there are an infinite amount of 'prime numbers' at your disposal, then there are an infinite amount of pairs.

NOW. The article started off describing primes separated by '2'. Then it goes on to discuss primes separated by larger (Much larger) amounts.

IS the original problem about primes separated by '2', or primes separated by any random number?

I don't see the problem being quickly unraveled if it's the latter.

[cartan]

I would suggest that if you stipulate that there are an infinite amount of 'prime numbers' at your disposal, then there are an infinite amount of pairs.

It is quite easy to see that there is an inifinitude of prime numbers: If you assume that there are only finitely many primes, and p₁, …, p_n are all prime numbers, then the product

p₁×…×p_n + 1

will be divisible by neither of these primes, and is bigger than any of the others, which leads to a contradiction! Hence, there are infinitely many primes. Unfortunately, this argument says absolutely nothing about the question whether there are infinitely many primes that are seperated by only 2 from another prime number!

NOW. The article started off describing primes separated by '2'. Then it goes on to discuss primes separated by larger (Much larger) amounts.

IS the original problem about primes separated by '2', or primes separated by any random number?

I don't see the problem being quickly unraveled if it's the latter.

The original problem is about primes seperated by 2. Proving it about primes being seperated by a little more is evidently simpler (but still very very hard!).

[Dalberg-Acton]

The problem is proving it to infinity. Unless I misunderstand what the word “prove” means. Some things like 1/3 or 0.33333, I assume repeats infinitely but that's not the same as proving the prime pairs is it?