Saturday, May 29, 2004

A Proof of the Twin-Prime Conjecture?

A Slashdot post pointed me to this purported proof by Vanderbilt University's Richard Arenstorf of the Twin-Prime Conjecture, which states that there are infinitely many primes of the form P, P+2, e.g, 5 and 7, 11 and 13, 17 and 19, or 29 and 31. This supposed proof, if correct, would be one of the most exciting results in number theory in quite some time.

I'm not and never have been an analytic number theorist, and I can't say that I've ever heard of Mr. Arenstorf before today, so I'm in no position to assess the likelihood that his proof isn't just stuff and nonsense, at least not without getting a chance to actually read it through. Going from the comments of some of the Slashdot readers, this guy got his PhD 48 years ago, and hit his productive prime in the 1960s; prior to this, his most recent published paper was in 1993. This hardly makes one feel confident in the correctness of his work, as mathematics is notorious for being a young man's game, the likes of Paul Erdos aside. As another commenter noted, I'd feel far more confident that this guy's hit the mark if the likes of Peter Sarnak, Andrew Granville, Andrew Odlyzko or Preda Mihailescu were to give his paper a preliminary thumbs up - that would still be no guarantee that Arenstorf was correct, of course, but it would lessen the odds of his being wrong considerably.

One thing I noticed in the abstract of the paper was that Arenstorf claimed that his proof depended only on methods from classical analytic number theory, and a quick scan through suggests that this is indeed the case, as it seems to lean most heavily on the theory of generating functions. This leads me to harbor some hope that I really will be able to read through the whole thing if I have the requisite patience - now all(!) I have to do is find the necessary free time!