[math-fun] Twin prime proof offered
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
At 06:56 PM 5/28/2004, Richard Schroeppel wrote:
Title: There Are Infinitely Many Prime Twins Authors: R. F. Arenstorf http://arXiv.org/abs/math/0405509
The proof was withdrawn yesterday. The page now reads: "A serious error has been found in the paper, specifically, Lemma 8 is incorrect." Also, although the original work was mentioned on math-fun in March 2003, I don't think anyone pointed out that an error was found in Goldston and Yildirim's attempt to prove that there are infinitely many "small" prime gaps. Keith Devlin has a non-technical article describing how the error was found: http://www.maa.org/devlin/devlin_06_03.html . It looks like the Twin Prime Conjecture won't give up without a struggle. -- Fred W. Helenius <fredh@ix.netcom.com>
Keith Devlin has a non-technical article describing how the error was found: http://www.maa.org/devlin/devlin_06_03.html .
Math-funsters of more genteel sensibility should perhaps be advised that the cited page is also yet another inflammatory political polemic disguised as something else, in this case popular mathematical exposition. Though I'm quite interested in both the specific (twin prime) and general (proof) topics, Devlin's ill-advised--and wholly inaccurate--forced "right-wing versus left-wing" metaphor rendered the screed ultimately unreadable. Statements such as "And there you have the problem. Like right wing policies, for all that it appeals to individuals who crave certitude in life, the right wing definition of mathematical proof is an unrealistic ideal that does not survive the first contact with the real world. (Unless you have an army to impose it with force, an approach that mathematicians have hitherto shied away from.)" belie at best a tragic authorial incapacity to forbear freighting an arguably interesting essay with his irrelevant agenda. How can anyone so educated possibly mistake these (sadly all too prevalent) sort of incontinent digs for civilized intelligent discourse? Enough with the obsessive bias already. Please note that I have no intention of starting a political flame-war on this list, and have refrained from debating the specifics of Devlin's position. I just object to this kind of journalism. So, invitiating venting aside, I think it's merely fair play to balance his pulpit with this one obscure small voice in protest. Thanks as always for your estimable indulgence, Your Admiring Correspondent.
Apart from an irrelevant appeal to the left wing political sentiments common among mathematicians, Devlin has expressed a point of view about proofs held by quite a few mathematicians and even more prevalent among philosophers of mathematics. The main pioneer of this view was Imre Lakatos in his book _Proofs and Refutations_. The book concerned proofs of Euler's formula E + 2 = F + V for the number of edges, faces and vertices of a plane polygon. My opinion is that Euler's proof was correct though informal. I suppose my own view would be characterized by Devlin as even more right wing than any he mentions. My solution to the difficulty of verifying very long and complicated proofs is to check them with an interactive computer prover and verifier. If computer programs are used to check some of the conjectures, then these programs need computer verified proofs that they meet their specifications. However, the state of the art of interactive theorem proving needs great improvement before it is practical for hard theorems. It seems to me that no great theoretical advance is needed, just a lot of hard work. I'm not well acquainted with the state of the art, but one landmark for me was Shankar's proof of the first Gödel incompleteness theorem using the Boyer-Moore interactive prover ACL2. One of their students gave a 50 step verified proof that a checkerboard with two diagonally opposite corner squares removed cannot be covered by dominoes, each covering two adjacent squares. I gave a four step version of the usual informal proof using set theory. Alas, there is no verifier that can accept my proof. I conjecture that until there is such a verifier, interactive provers will not be used by mathematicians for difficult theorems. I used Jussi Ketonen's interactive prover EKL for first order logic in class, and the students used it successfully to prove easy theorems about Lisp functions. I did not have the energy or resources to carry the project further. I see the major step toward use of interactive provers in mathematics as being the creation of what I call heavy-duty set theory (HDST). ZFC, is normally axiomatized using a minimal set of axioms, because the main use of the formalization has been to prove metatheorems, e.g. the independence of the continuum hypothesis about ZFC rather than to prove theorems within ZFC. HDST should have a large number of theorems of ZFC as axioms and admit verification in a single steps inferences that are automatically verifiable. Once the prover itself has been proved correct and the proof verified, we can have confidence in proofs produced and verified interactively using it. Note that proving the prover correct is a single mathematical task, and it is unlikely to be anywhere near as onerous as checking candidate proofs of the twin primes conjecture. Proving correct the verifier, the compiler for the language in which the verifier is written, and the machine used are a small number of small tasks that can be done once.
participants (4)
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Fred W. Helenius -
John McCarthy -
Marc LeBrun -
Richard Schroeppel