bs@faron.UUCP (Robert D. Silverman) (12/27/84)
For those of you who haven't heard yet, the Riemann Hypothesis has apparantly been proved by a mathematical physicist at the University of Paris. His name is Matsumoto (sp?). The proof has been seen by others and appears to be correct. It is also likely that the Generalized Riemann Hypothesis has been proved along with it. For those of you who are unfamiliar with the problem it is considered to be (or was) the most outstanding problem in all of mathematics. The Zeta function is an analytic function of z = a+bi in the half plane a > 1 and can be extended so that it is analytic everywhere except at z = 1 where it has a first order pole. The hypothesis states that all of its zeros lie on the line Real(z) = 1/2 and has many implications in the theory of prime numbers. A good reference for the problem may be found in (among others) "Topics from the Theory of Numbers", Emil Grosswald, MacMillan Co., 1966. One very important aspect of the problem is that if the generalized hypothesis it true there exists a deterministic, polynomial time algorithm for proving numbers prime. I do not know the details of the proof except in a very vague way. Apparantly there is a class of operators relating to the Zeta function whose positive definiteness is equivalent to the hypothesis. Matsumoto has shown that those operators are in fact positive definite.
chongo@nsc.UUCP (Landon Noll) (08/15/85)
Around the end of 1984, I had heard that someone was doing some final touches on what they claimed was the resolving of the Riemann Hypothesis. Does anyone know the status of the paper? It is still pending, or has a flaw been found in the argument? chongo <> /\../\ -- no comment is a comment.
cjh@petsd.UUCP (Chris Henrich) (08/17/85)
[] About the end of 1984, someone did announce a proof of the Fermat conjecture (a/k/a Fermat's Last Theorem). The proof was to be delivered at the annual meeting of the American Math Society, but because the speaker was ill this did not happen. There were things about the announcement that made some mathematicians feel it was unlikely to be valid. Since nobody has seen the proof, the question remains unsettled. Rumors of a proof of the Riemann Hypothesis have been circulating, especially in noisy environments like Usenet. But I have seen no notice in such places as _Scientific_American, _The_Mathematical_Intelligencer_, or _The_Notices_of_the_American_Mathematical_Society_. I really think somebody must have confused the two conjectures. Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 758-7288
bill@milford.UUCP (bill) (08/20/85)
Has anyone seen the well publicized proof by Falting of Mordell's Conjecture? This was supposed to be very close to establishing Fermat's Theorem, but I haven't heard much about it recently.
russ@hao.UUCP (Russell K. Rew) (08/21/85)
> Around the end of 1984, I had heard that someone was doing some final touches > on what they claimed was the resolving of the Riemann Hypothesis. Does > anyone know the status of the paper? It is still pending, or has a flaw > been found in the argument? > > chongo <> /\../\ Although I don't have the reference handy, there was an assertion that the Riemann hypothesis had recently been proved in one of the November or December 1984 issues of The New Scientist (a respectable British weekly periodical similar to but more comprehensive and opinionated than Science News). This appeared in a cover story "1984: The Golden Age of Mathematics," in which 1984 was proclaimed a great year for mathematics because three longstanding open questions were finally resolved: the Riemann hypothesis, Bieberbach's conjecture, and Merten's (sp?) conjecture. I remember that the article mentioned that the Riemann hypothesis had been proved by a Japanese mathematician in Paris, but that the proof had not yet been widely circulated, so that there was some skepticism about its validity. -- Russ Rew USENET: ...!hao!scd-sb!russ CSNET: russ@ncar
jwt@CS-Arthur (Jon W. Tanner) (08/22/85)
> Has anyone seen the well publicized proof by Falting of > Mordell's Conjecture? This was supposed to be very close > to establishing Fermat's Theorem, but I haven't heard much > about it recently. Gerd Faltings' proof of Mordell's conjecture has apparently been accepted by the mathematical community, though I haven't seen it. As a particular application of Mordell's conjecture, there can be at most finitely many solutions to the equation x^n + y^n = z^n in relatively prime integers x, y, and z, for a given n > 3. Fermat's "theorem" claims that there are *no* solutions.
bs@faron.UUCP (Robert D. Silverman) (08/22/85)
The Riemann hypothesis proof by Matsumoto was withdrawn. Bombieri, Selberg, and others found flaws in the proof that couldn't be patched. For those of you who are not acquainted with Merten's conjecture: It was recently disproved by some people at Bell Labs (Odlyzko and Lagarias I believe). It states that: SUM (u(x)) <= sqrt(N) where u(x) is the Mobius function x = 1,N defined as: 1 if x is squarefree and has an even number of distinct prime factors 0 if x is not squarefree -1 if x is squarefree and has an odd number of distinct prime factors. The truth of Merten's conjecture would have proved the Riemann Hypothesis. An assertion was made recently that the proof of Mordell's conjecture was 'close' to a proof of Fermat's Last Theorem. In fact, they are only very vaguely related. The truth of Mordell's conjecture simply establishes that for and GIVEN exponent N > 2: A^N + B^N = C^N has at most a finite number of solutions. It doesn't say anywhere that the number of solutions is zero. A closer result is a recent one of Adelman et.al. who proved that Fermat's Last Theorem was in fact true for an infinite number of exponents. This of course does not mean that it's true for ALL exponents. As I recall they did establish that the set of exponents for which the theorem was true has positive density but that the density was less than 1. Bob Silverman (they call me Mr. 9)
tracy@ihuxl.UUCP (Kim) (08/27/85)
<Mr. 9 (Bob Silverman)> ..... > A closer result is a recent one of Adelman et.al. who proved that > Fermat's Last Theorem was in fact true for an infinite number of exponents. > This of course does not mean that it's true for ALL exponents. As I recall > they did establish that the set of exponents for which the theorem was true > has positive density but that the density was less than 1. > Is that a positive density strictly less than 1 or less than or equal? Kim Tracy AT&T Bell Labs, Naperville,IL ..ihnp4!ihuxl!tracy
paulv@boring.UUCP (09/01/85)
In article <332@faron.UUCP> bs@faron.UUCP (Robert D. Silverman) writes: >The Riemann hypothesis proof by Matsumoto was withdrawn. Bombieri, Selberg, >and others found flaws in the proof that couldn't be patched. > >For those of you who are not acquainted with Merten's conjecture: It was >recently disproved by some people at Bell Labs (Odlyzko and Lagarias I >believe). > >The truth of Merten's conjecture would have proved the Riemann Hypothesis. > Mertens' conjecture was disproved by M.A. Odlyzko of AT&T Bell Labs and H. te Riele of the Centre for Mathematics and Computer Science (CWI) in Amsterdam. The (dis)proof has appeared in Mathematische Annalen (?) See also M.A. Odlyzko, A Disproof of the Riemann Hypothesis, in: Dopo le Parole (Lenstra^2 & v.Emde Boas, eds), Amsterdam, 1984 for some other results.
rpb@aaec.OZ (Bob Backstrom) (09/05/85)
> Gerd Faltings' proof of Mordell's conjecture has apparently been > accepted by the mathematical community, though I haven't seen it. As a > particular application of Mordell's conjecture, there can be at most > finitely many solutions to the equation x^n + y^n = z^n in relatively > prime integers x, y, and z, for a given n > 3. Fermat's "theorem" > claims that there are *no* solutions. [ Another posting to the net since mail to jwt@CS-Arthur.ARPA was returned with Host unknown message. ] It's amazing how close Fermat's Theorem actually comes to being false. Consider the following sums of cubes: 3 3 3 6 + 8 = 9 - 1 3 3 3 71 + 138 = 144 - 1 3 3 3 135 + 138 = 172 - 1 3 3 3 17328 + 27630 = 29737 - 1 etc. etc. as well as sums with + 1 on the r.h.s.: 3 3 3 577 + 2304 = 2316 + 1 3 3 3 13294 + 19386 = 21279 + 1 etc. There are infinitely many such occurrences as can be shown by the following three algebraic identities: 3 3 4 3 4 3 (1) (9n + 1) + (9n ) = (9n + 3n) + 1 3 3 4 3 4 3 (2) (9n - 1) + (9n - 3n) = (9n ) - 1 2 3 3 3 3 3 (3) (6n ) + (6n - 1) = (6n + 1) - 2 Not all of the above examples can be generated from formulas (1) and (2), so the problem is to find ALL the "near" Fermat solutions. From Bob Backstrom, Australian Atomic Energy Commission, Sydney, New South Wales, Australia.