I just want to know if there are infinite twin primes. It is one of those things that are easy to formulate but very hard to prove. That is probably why I like it. An idiot like me can understand it but because it is unsolved I get to pretend that I am doing something special.

Twin primes aren't actually very special, they are just one case of a prime gap.

p_(n+1) - p_n = m

Where m = 2

But you could set m to any positive integer and ask questions about the distribution of gaps that size.

For m=1 there is only 1 pair at n=1. {2,3}

Then all odd m have 0 occurrences.

But if you pick an even m there appear to be infinite but it is a bear to prove. I think real mathematicians have proven that at least some small gap sizes must occur infinitely often. I think the best we know is that at least one gap size less than or equal to 246 occurs infinitely often.

The best I have done myself, was to show that for any finite list of primes there are infinitely many twins co-prime to the numbers in the list, and I can even count them for you. The problem is that isn't a finite list of primes.

The problem has had me nerd-sniped for over 20 years because it is just so unlikely that there could be a finite number of them. If given some finite list of primes I can always find a huge number of candidate twin primes, then they would all have to be supremely unlucky to end up cancelled by some larger prime with none escaping ever.