Wikipedia, List of common misconceptions, here. via Big Picture
Christian Perfect, The Aperiodical, Bound on prime gaps decreasing by leaps and bounds, here. This feels a little bit like Shelby Lyman broadcasting Fischer Spassky. I like it. Fischer still has google name search pole position.
The bound quickly tumbled much further than that, and Terry Tao initiated a new Polymath project to coordinate efforts. Participants are trying to optimise three constants:
- H is a quantity such that there are infinitely many pairs of consecutive primes of distance at most H apart. Would like to be as small as possible.
- k0 is a quantity such that every admissible k0-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible.
- ϖ is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in ϖ lead to improvements in k0.
Records are being kept on the Polymath wiki, and progress is swift: just yesterday eighteen different values for H were recorded, along with four values of k0. As I write, they currently think that H≤108,540. Of course, in the rough and tumble of mathematicians trading arguments through comments pages it’s entirely possible that mistakes will go unnoticed, but it’s certainly fun to watch!
Interested followers should take a look at an online reading seminar conducted by Terry Tao which is going through the original paper.