Daniel Wigton on Nostr: I just want to know if there are infinite twin primes. It is one of those things that ...
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.
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.