20040528, 19:25  #1 
"Bob Silverman"
Nov 2003
North of Boston
2^{2}×3×7×89 Posts 
Twin Primes Conjecture
In case noone has heard, Richard Arenstorf (emeritus, Vanderbilt) has
published a manuscript with a proof of the twin prime conjecture. It is too early to say whether the paper might contain errors (I have skimmed it and probably lack the skills to determine any subtle errors). However, the basic approach looks sound to me and represents some serious analytic number theory. Basically: Let L(n) be the VonMangoldt function. Arenstorf considers the Dirichlet series T(s) = sum(n > 3) L(n1)L(n+1)n^s for Re(s) > 1 and shows that T(s) B/(s1) has an analytic continuation onto Re(s) = 1. This allows use of a Tauberian Thm to show that sum [log(p) log(p+2) > oo. Indeed, it allows showing that the density of twin primes (the constant B) matches the conjecture of Hardy & Littlewood. I do not know the details of the Tauberian Thm, nor how it is applied. [note for the uninitiated. A Tauberian Thm allows one to deduce local behavior of a function from its long term ergodic behavior] I will be trying to read the paper in depth this weekend. I will need to do so with references at my side. This is a SERIOUS piece of work. 
20040528, 19:34  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
3^{2}·1,193 Posts 
In a nut shell does it say that twin primes are endless, or what?

20040528, 19:45  #3  
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Quote:
(say WHAT??? ) But seriously, this is great news and'd put one of the most famous questions in number theory to rest. The paper can be found here. I'll skim over it but I don't expect to understand anything beyond the four lines lines of the abstract. Perhaps you'd like to give a very general summary, in layman's terms, of proof method used when you read it? Alex 

20040528, 19:54  #4  
"Bob Silverman"
Nov 2003
North of Boston
16464_{8} Posts 
Quote:
know very much about Tauberian methods (except in a general way). They allow one to deduce *local*, finite behavior of a function from its ergodic behavior. Setting the problem up as a Dirichlet series is quite clear; this is how one proves there are infinitely many primes in any arithmetic progression. The use of VonMangoldt's function in the series is quite clever (but may have been obvious to one more skilled than I). I will try to summarize after I have read it. I am not an expert in analytic number theory. I am perhaps at the level of the average grad student in the subject. Bob 

20040528, 23:37  #5  
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Quote:
Unfortunately, I can't understand much more... Using a Google search on Arenstorf +"Twin Prime Conjecture" returned only one result:http://www.math.vanderbilt.edu/~cale.../98_11_09.html Last fiddled with by jinydu on 20040528 at 23:50 

20040529, 00:12  #6 
∂^{2}ω=0
Sep 2002
República de California
11,743 Posts 
If true, this would be a HUGE result. Thanks for the headsup, Bob, and thanks for the link to the manuscript, Alex.

20040529, 01:09  #7 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
3^{2}×1,193 Posts 
should we search for the twins of Mersennes?
We could bag the largest twins. Is P95 capable of doing that? Or G or M lucas? Does the math work to do that (check for a twin of the M's)? 
20040529, 01:28  #8 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Hmmm, the largest twin prime pair is tiny, only 51090 digits!

20040530, 17:21  #9 
Banned
"Luigi"
Aug 2002
Team Italia
29×167 Posts 
If there are infinitely many, it's only a mater of time to find them out
Luigi 
20040531, 02:23  #10 
∂^{2}ω=0
Sep 2002
República de California
11,743 Posts 
On a related note, I was perusing the 21. May issue of Science this afternoon, and came across the following article in the News of the Week section, with the impressively alliterative title "Proof Promises Progress in Prime Progressions":
http://physicalsciences.ucsd.edu/new...eory052404.htm 
20040531, 02:33  #11 
Dec 2003
Hopefully Near M48
2·3·293 Posts 

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