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Math Science

Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof 172

Bryan writes "Xian-Jin Li's purported proof of the Riemann Hypothesis (reported on recently) has been rebuked by Fields Medalist Terence Tao. Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts." Relatedly, jim.shilliday writes "The proof cites and appears to be based in part on the work of the leading French theorist Alain Connes. A few hours ago, Connes posted a comment on his blog stating that the purported proof is so badly flawed that he stopped reading it."
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Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof

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  • Why "fortunately"? (Score:5, Interesting)

    by fgaliegue ( 1137441 ) on Friday July 04, 2008 @01:36PM (#24060721)

    From the summary:

    Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts

    Why? I'm probably missing something obvious, I'm not even a mathematician to start with, but...

    I mean, we (the world) do want to prove it right (or wrong) one day or another, don't we?

    • by Anonymous Coward on Friday July 04, 2008 @01:39PM (#24060761)

      I guess they mean that there's no shame in having failed, since many other respectable attempts also failed.

      • Re: (Score:2, Informative)

        by Anonymous Coward

        Li did respectable work once and has made a large faux pas in his handling of this affair, but it is now over. Let's focus on something far more interesting if we're talking about the Riemann Hypothesis - a wonderful (translation of a) transcript of an interview with Atle Selberg [ams.org], which makes fascinating reading.

    • by Chris Pimlott ( 16212 ) on Friday July 04, 2008 @01:43PM (#24060787)

      They're just being polite by pointing out there's no shame in failing to prove the Riemann Hypothesis, since it has frustrated the attempts of many a prominent mathematician so far.

      • Re: (Score:3, Interesting)

        by sohare ( 1032056 )
        I'm not so sure they're being polite. Mathematics has it's share of cranks, and high profile conjectures receive a lot of attention by these woomeisters. While a crackpot proof might appear mystical to the layperson due to the extreme use of technical jargon, a trained mathematician can usually spot a uninformed line of argument. To draw a more comprehensible analogy, I would liken many of the proofs of longstanding problems to the endless stream of perpetual motion machine patents. Except the latter de
        • by grizdog ( 1224414 ) on Friday July 04, 2008 @03:10PM (#24061465) Homepage
          First of all, while you are right that there is no shame in failing to prove the RH, there is some shame involved in announcing in such a high-profile way that you have done it, and effectively requiring everyone to stop what they are doing to read your proof.

          Having said that, Li is no crank. I had not heard of him, but that's no surprise since I'm not a number theorist. But he has published several refereed papers in this area, has a position at BYU, and really ought to have known better than to explode on the scene like this.

          I've gotten communications from genuine crackpots, wanting my comments on their work. Early in my career, I wrote back, gently pointing out the mistake. To my horror, friends then received slightly modified but still absurd drafts, listing me as a collaborator! Li is a real mathematician, probably with poor social skills, and a bad proof.

          • by Jerf ( 17166 ) on Friday July 04, 2008 @09:51PM (#24063707) Journal

            announcing in such a high-profile way

            Are you sure about that? Getting a paper onto arxiv.org doesn't seem to be that hard, and there's lots of ways to find out about it (RSS feed, etc.). He may not have had any reason to believe that he'd get this sort of attention, as he may have thought everyone involved would simply assume that it wasn't worth much, not having been peer reviewed.

            While I love the free and open flow of information that arxiv represents, this is hardly the first time that something has been posted on there and subsequently blown out of proportion. The Internet at large doesn't seem to really understand arxiv.org, that just because someone's got a fancy LaTeX paper up claiming some wild thing doesn't mean it's credible. A paper on arxiv.org shouldn't even be understood as being endorsed by the author, let alone "science". I always love when somebody backs up their argument about physics with a link to arxiv.org, it's like a red flag that it's time to just pack it in, you're not going to get through to this person, because they only understand the trappings of science, not the actual process.

    • by Ardeaem ( 625311 )
      My understanding is that if a proof is found, it has the potential to lead to the undermining of current encryption methods, which depend on the difficulty of factoring large prime numbers. This would be disastrous for online commerce and security.
      • by FnH ( 137981 ) on Friday July 04, 2008 @03:38PM (#24061703)

        I believe you're mixing this up with another hard problem that hasn't been proven yet. You're thinking about the NP = P [wikipedia.org] problem. The difference is that here we don't know what will be the outcome, whereas for the RH most people assume it's true. Having a proof for this wouldn't really change anything (apart from validating large parts of mathematics that assume it is true)

        • Re: (Score:3, Interesting)

          by Ardeaem ( 625311 )
          No, I wasn't saying that the RH being true would cause problems with encryption (I am aware that most mathematicians assume it is true), but rather that the methods used to prove it would cause complications. See here: link [it-director.com]
        • For the record, most people think P != NP.
          • Re: (Score:3, Insightful)

            by Anonymous Coward

            For the record, most people don't know what P or NP is ;)

      • by Rudolf ( 43885 ) on Friday July 04, 2008 @04:13PM (#24061971)

        [..] lead to the undermining of current encryption methods, which depend on the difficulty of factoring large prime numbers.

        That's a trivial problem.

        All prime numbers have two factors: 1 and itself.

        Goodbye encryption :-)

      • by gomoX ( 618462 ) on Saturday July 05, 2008 @02:35AM (#24064599) Homepage

        One possible explanation for your understanding (which in my understanding, is wrong), is the Miller-Rabin primality test algorithm.

        The primality problem (telling whether a number is prime), although hard, was never proved to be NP-complete.
        The Miller-Rabin primality test is a (actually, the 1st and possibly the only) polynomial deterministic algorithm that is based on the Riemann hypothesis (polinomial deterministic meaning "fast and accurate"). Proving RH would prove that Miller-Rabin is exact and therefore shown that primality testing is in P.

        http://en.wikipedia.org/wiki/Miller-Rabin_primality_test [wikipedia.org]

        Unfortunately, algorithm freaks were faster than math freaks (well, the algorithm freaks involved were math freaks too) and a new algorithm called AKS was developed that did everything Miller-Rabin did without relying on the Riemann Hypothesis.

        http://en.wikipedia.org/wiki/AKS_primality_test [wikipedia.org]

        So, to this day, we know primality testing is polynomial. The _real_ problem in cryptography is prime *factoring* (if it's not prime, then find 2 numbers that when multiplied produce the original number). Although it is not know whether that problem is P or NP-complete or both, it is believed to be outside NP because it is much harder than plain primality testing.

        http://en.wikipedia.org/wiki/Integer_factorization [wikipedia.org]

        • by makomk ( 752139 )
          "Although it is not know whether that problem is P or NP-complete or both, it is believed to be outside NP because it is much harder than plain primality testing."

          I think you mean that it's outside P. It's obviously in NP, since we can verify any possible factorisation in polynomial time.
  • I have to ask, I know for mathematicians this is a big deal and all, but what are the piratical applications for this?
    • by HappySmileMan ( 1088123 ) on Friday July 04, 2008 @01:42PM (#24060777)
      Well it doesn't have any piratical applications, but the ninjas will definitely find a use for it
    • by the eric conspiracy ( 20178 ) * on Friday July 04, 2008 @01:56PM (#24060889)

      There are a lot of results based on assuming the conjecture is true, including a variety of factoring and root finding algorithms that are computationally very useful.

      Until it is proven you really don't know if these algorithms are giving correct answers.

      This is why it is so important and has a big prize associated to it.

      • by Dr. Blue ( 63477 )

        The conjecture doesn't affect whether the answers are correct - that's easy enough to verify (even, despite several follow-ups, if the algorithm claims values/factors are prime - even before it was shown that primality testing is in P, there were known ways to prove a value was prime, showing that PRIMES is in NP).

        However, it does affect the running time analysis. There are several algorithms that have a run-time analysis that says something like "Assuming the Riemann Hypothesis, the running time is ...."

        S

    • I had to go look it up after you asked... apparently to mathematicians, there are plenty of practical applications. See http://en.wikipedia.org/wiki/Riemann_hypothesis [wikipedia.org] for a few examples. Neither of them made sense to me yet, and I've already had coffee. If they are looking for one example where this theory is not true, and offering a million bucks, someone is sure to put a couple yellow dog games console clusters together and find out soon enough. (either that or prove Doom is written by zombies who don't

  • by pongo000 ( 97357 ) on Friday July 04, 2008 @01:37PM (#24060729)

    The "proof" is that of Theorem 7.3 page 29 in Li's paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).

    ...but this certainly cleared things up for me!

    • The "proof" is that of Theorem 7.3 page 29 in Li's paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).

      ...but this certainly cleared things up for me!

      ...this brings me back so many memories from my old grad days... the books usually brought a demonstration of a simple theorem , and the exercises were to demonstrate theorems WAY harder, stuff like " Using the theorem that states that 0+1=1, prove that if God exists he doesn't want you to finish College." =)

      • by Beardo the Bearded ( 321478 ) on Friday July 04, 2008 @02:09PM (#24060975)

        It's called "Proof by Intimidation":

        using the formula:

        [ some formula ]

        it is trivial to see that:

        [ some other formula out of nowhere ]

        therefore, combining the above, we can arrive at the easily obtained answer:

        [ some MATLAB result ]

        Don't forget, it works both ways; the people marking your assignment don't want to admit that they can't see the so-called "trivial" derivation.

        • by ceoyoyo ( 59147 )

          I know a paper that uses that approach. "And it is obvious that the inverse operation may be be accomplished by departitioning the spectrum...."

          A colleague and I actually looked into the "obvious" problem and found some pretty powerful results that the author of that paper is going to be kicking himself for missing.

    • by Beardo the Bearded ( 321478 ) on Friday July 04, 2008 @02:15PM (#24061009)

      Use the "Star Trek" filter:

      he is extending the test function h from [ tech ] to [ tech2 ] by [ tech 3 ] and then using Fourier transform ... This cannot work and [ tech ] form a set of measure 0 [ tech 4 ] (unlike what happens when one only deals with finitely many places).

      When he moved from one set to another and did the Fourier transform, he forgot that he ended up with an empty set instead of a finite number of points because that's apparently a property of whatever the hell he was talking about.

    • Re: (Score:2, Insightful)

      by Larsrc ( 1285062 )

      No shit. I took a minor in math, really loved and grokked things, up to a certain level. Beyond that I suddenly have no fscking clue what they're even talking about. When looking at similar levels of, say, biology, I at least have a faint idea of what it's about. High-level math is weird.

    • Re: (Score:3, Interesting)

      This is not all that bad.

      Probably many slashdotters are familiar with the discrete Fourier transform (used in JPEG encoding, incidentally). The DFT for sequences of length n fits together nicely with the DFT for longer sequences whose length is a multiple of n. If you try to put all these DFTs for sequences of different length together in a certain way and combine them with the continuous Fourier transform, you end up with something called the adelic Fourier transform. (That's a little bit different

  • Ow my head (Score:5, Funny)

    by Jailbrekr ( 73837 ) <jailbrekr@digitaladdiction.net> on Friday July 04, 2008 @01:38PM (#24060745) Homepage

    The proof, and the rebuke, only proved my theory that there is a distinct surge in advil usage when something like this is posted on /. or digged.

    • I stopped reading when I saw that he is using Advil, this cannot work when it is well established that Excedrin is the preferred migraine reliever.

  • by the eric conspiracy ( 20178 ) * on Friday July 04, 2008 @01:38PM (#24060747)

    Well duh this is what we have been saying - this is a preprint and is likely to have errors. Whether or not they can be repaired is open to question.

    Wiles' proof of Fermat's last theorem took a long time to go through the review and repair process. And there was at least one pretty hard problem that had to be fixed.

    Slashdot's "journalistic" process really suxors when it comes to this sort of stuff.

    • by proverbialcow ( 177020 ) on Friday July 04, 2008 @01:45PM (#24060811) Journal

      Slashdot's "journalistic" process really suxors when it comes to this sort of stuff.

      Wel of course it does. Slashdot is journalology, not journalonomy.

    • Re: (Score:3, Interesting)

      The comments made by Tao and Connes are the sort of comments one would make if the paper was irrevocably flawed. For instance, Tao notes that "the decomposition claimed in equation (6.9) ... is, in fact, impossible; it would endow the function h ... with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry ..."

      In more simple terms: Partway into the paper the author proved something that is definitely false; he then relied on this false

    • by pavon ( 30274 )

      Yeah, this is becoming a real problem with the preprint journals. Media groups like New Scientist will run a hyped-up story on some "ground-breaking new development" which will have propagated through the blog echo-chamber before other scientists have even had a chance to review it. It's not enough for the media to completely butcher the science they do present, now they have to present results which haven't even had cursory review. It's no wonder the public doesn't trust science considering what is is bein

  • by ActusReus ( 1162583 ) on Friday July 04, 2008 @01:42PM (#24060781)
    Oh come on, you were almost there! How about:

    "Renowned Researchers Rebuke Recent Riemann Reasoning"
  • The "proof" contained so many errors that a previously uncontacted amazon tribe could have debunked it.
  • What a shock... (Score:5, Interesting)

    by porcupine8 ( 816071 ) on Friday July 04, 2008 @02:04PM (#24060939) Journal
    My husband is a mathematician, and he gets emails weekly from crackpots claiming to have disproved the proof of Fermat's Last Theorem or having proven the Riemann hypothesis or whatever. You can submit anything to the ArXiv, this shouldn't have even been news in the first place until it was confirmed.
    • Re:What a shock... (Score:5, Interesting)

      by njj ( 133128 ) on Friday July 04, 2008 @03:39PM (#24061709)

      I work part-time for a couple of mathematics research journals and we do get the occasional crank submission. There's one guy who's been sending us, on average, a 'paper' every week or so for the past few years: typically a single, badly-written page of gibberish (we're talking Time Cube [wikipedia.org] standard lunacy here) which is clearly not the work of someone who's ever seen a real mathematics paper. We've never responded to him, or even acknowledged any of his submissions (helpfully he prints his return address on the back of the envelope, so these days they go straight in the bin, unopened and unread) and yet he still keeps sending them in.

      The arXiv also tends to get its fair share of crank submissions, usually elementary attempted (but trivially broken) proofs of things like the Goldbach Conjecture, Fermat's Last Theorem and the like - I'm assuming that the really mad stuff is filtered out by the moderators.

      In contrast, at a quick glance to my nonspecialist eyes (I'm a knot theorist) Xian-Jin Li's preprint looks like a genuine (if flawed) attempt by a serious, qualified mathematician who specialises in the relevant area. Fair play to him for trying, though. I'm also not sure I'd characterise Terence Tao or Alain Connes' refutations as 'rebukes' - they looked more like dispassionate analyses of the paper's flaws to me, the sort of discussion you'd expect from the peer-refereeing process.

      • (I'm a knot theorist)

        So how is work coming on the time cube knot ?

        • IANAM, but I love to read about it. Maybe you can answer this for me, since you are a knot expert. ;-)

          I've read that you can't have knots in 4-space, so how do 4-dimensional beings tie their shoes?

          • I've read that you can't have knots in 4-space, so how do 4-dimensional beings tie their shoes?

            Velcro, but unfortunately there are eight pieces to a velcro set there. (The proof of this is, of course, trivial, per novakyu (636495)). All sentient creatures in 4-space must therefore have at least a pair of hyper-thumbs (thumbs cubed), and not just supra-thumbs (thumbs squared), QED.

      • by Gnavpot ( 708731 )

        helpfully he prints his return address on the back of the envelope, so these days they go straight in the bin, unopened and unread

        This person is obviously a lawyer. He is preparing you for the real and very important letter that he is obliged by law to send you but really does not want you to read before it is too late.

  • by HuguesT ( 84078 ) on Friday July 04, 2008 @03:52PM (#24061801)

    Just wanted to point out that Professor Connes is also a Fields medalist (1982) [wikipedia.org].

    I guess it is a testament to Xian-Jin Li excellent reputation and the importance of the topic that these two mathematical superstars took the time to look at his proof.

  • by phr1 ( 211689 ) on Friday July 04, 2008 @04:10PM (#24061931)
    And the slashdot post I think miscasts Connes's remark. It's not like Connes quit reading the proof because it so full of crap that Connes got disgusted. Proofs are chains of reasoning that don't hold together if there is a single link that's flawed. So as soon as Connes found an error that he didn't see how to fix, there wasn't any point to continuing, everything that relied on the erroneous step simply couldn't be supported. Like if I tell you my plan for making a 1000 mpg car, and it turns out to depend fundamentally on steel being lighter than air. This dependence might be subtle enough that neither of us realized it at first, so I'm not necessarily a crackpot for coming up with such a plan. But as soon as the problem is noticed, the rest of the details become irrelevant.

    The proof was a legitimate effort by a non-crackpot, but the ideas in it were well known to specialists in the field and were generally understood to not be powerful enough to crack the problem. So the errors were found fairly quickly. Scott Aaronson's post Ten Signs that a claimed mathematical breakthrough is wrong [scottaaronson.com] item #10 may be helpful in understanding what happened.

    • Like if I tell you my plan for making a 1000 mpg car, and it turns out to depend fundamentally on steel being lighter than air. This dependence might be subtle enough that neither of us realized it at first, so I'm not necessarily a crackpot for coming up with such a plan.

      I thought i was the only one who made those mundane mistakes.

    • Re: (Score:3, Interesting)

      by khallow ( 566160 )
      A failed proof can still be worth reading, if it has interesting proof techniques or novel math structures in it. For example, ring theory, algebraic geometry, and moduli spaces were (as I understand it) due in part to failed proof attempts for Fermat's Last Theorem.
    • by Raenex ( 947668 )

      Like if I tell you my plan for making a 1000 mpg car, and it turns out to depend fundamentally on steel being lighter than air. This dependence might be subtle enough that neither of us realized it at first, so I'm not necessarily a crackpot for coming up with such a plan.

      Your post makes a good point, but this is the worst car analogy I've ever seen on Slashdot :)

  • "Rebuke"is not a technical word [google.com] and is certainly not what happened.
    The whole thing was a bit more polite than the way it has been descibed here - as anyone who follows the link will find, at least.
  • I think Connes was refuting the proof which is a more neutral term than rebuke (which means telling off).

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