Title: There Are Infinitely Many Prime Twins Authors: R. F. Arenstorf http://arXiv.org/abs/math/0405509 http://www.math.vanderbilt.edu/faculty/Arenstorf.html Richard Arenstorf is a real mathematician, at Vanderbilt. I looked at the 38 page manuscript - it's a well presented bunch of analytic number theory. The proof requires a lot of multi-level series manipulations, and careful arguments about convergence. I'm way too rusty to check this, but it's clearly a serious attempt. He notes that he's been working on the problem intermittently for 20 years. A Google search turned up a few references to his other work. The author claims a bit more: That the expected asymptotic expression for the count is correct, including the twin-prime constant. My quick scan didn't show any reason the proof wouldn't adapt to more general {P,P+2k} twins, although "Twin" would have to be agnostic about the primality of the intermediate numbers between P+2 and P+2k-2 when k>2. Similar ideas might also show the infinity of Sophie Germain pairs P,Q=2P+1 with P and Q both prime; and a restriction to P=3(mod4) would show an infinity of Mersenne composites Mp. The more general problem of primes in arithmetic progressions (longer than 2 of course) was recently settled by a density argument. Arbitrary lengths exist. I'd welcome comments from someone who remembers more of their analytic number theory than I do. Rich rcs@cs.arizona.edu