Twin Primes and Polignac's Conjecture
Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.
I've not posted here before but have been an interested reader. I wanted to run an attempted proof of the Twin Prime Conjecture (and perhaps the Polignac Conjecture in general) past the forum members - I know prime numbers tend to attract amateurs and cranks, and that most attempted proofs are flawed. But I'm optimistic enough about this attempt to try to get some feedback.
It's related to the infinite product fraction Euler inverted to show that the reciprocals of the primes diverge. There's a fairly brief summary at the start, then more detail beneath. If you do have time to take a look I'd be extremely grateful.
(It also has some similarity to the attempt by Alan Tyte discussed in an earlier thread - but where that one relies on a dubious averaging step, this one takes the observation of patterns a bit further and is hopefully more rigorous as a result).
* indicates original appearance(s) of problem.