Riemann Hypothesis Proved? 454
Theodore Logan writes "Has the Riemann Hypothesis finally been proved? The proof is a couple of months old, and to the best of my knowledge a Swedish newspaper is the only one to take up the story yet, so there is certainly a possibility that this is a hoax, or a less than watertight proof. But if it turns out to be the real thing, it will, apart from winning the authors eternal fame and glory for finding the holy Grail of modern math, provide them with a cool $1 million as they claim the first Millennium Prize." We had a story a while back about this as well.
Sooo... what is it exactly? (Score:1, Insightful)
The URL to the hypothesis is: http://www.utm.edu/research/primes/notes/rh.html
The URL contains the word "primes".
Primes are essential to much of todays cryptography, like public key encryption.
But what does the hypothesis say, in laymans terms?
What are the practical implications?
Anyone?
Re:Okay, assuming this proof to be correct... (Score:0, Insightful)
Re:Okay, assuming this proof to be correct... (Score:3, Insightful)
Who knows what use someone may derive from the proof of the RH?
Re:Okay, assuming this proof to be correct... (Score:3, Insightful)
Scanning Tunnelling microscopes are just one example. Based on the pure science of quantum mechanics, which was very easy to dismaiss as "of no practical use" for a good thirty years.
Re:Okay, assuming this proof to be correct... (Score:5, Insightful)
Not everything need have immediate application.
Re:You can help (Score:3, Insightful)
Re:Okay, assuming this proof to be correct... (Score:2, Insightful)
So mr Pytagoras you say that if one multiply the radius of a circle with two and then by approxymatley 3.14 one gets the size of that circle. Of what practical use is it again?? Anyone?
Re:Okay, assuming this proof to be correct... (Score:2, Insightful)
Re:You can help (Score:3, Insightful)
You might want to mention that to the people who finally proved the 4 colour conjecture a few years back then.
And anyway, even if you couldn't find a proof of this theorem through pure number crunching, you may be able to find a counter-example, which would be equally interesting.
What is it good for? (Score:3, Insightful)
How many prime numbers are there less than a given number.
It doesn't take much thought to work out why that would be handy in say cryptography.
But most complex maths starts for it's own sake. You build the tools in the knowledge that eventually someone is going to use them, and inevitably they always do.
I read about advances in nano-technology all the time. What's the point, no-one's using them? But without them now we wouldn't have cool stuff in 20 years. Same goes for maths. I would have thought nerds of all people would get that point.
Ok for the laymans (Score:2, Insightful)
Almost all "good" encryption uses prime numbers. If someone can figure out how to factor prime numbers, or find a quick way to determine a prime number, then all the fancy encryption in the world wont help because someone can just crack it in real time.
Right now public encryption works because it would take so long to break the encryption, even with say... 300k computers (distrubted.net) that when the encryption was broken (5-10 years later) the info would be old and it would not matter.
If someone could figure out some way to factor primes (which is along the lines of riemans sums) then they could possibly break even our stronger prime based encryption, which would make a great many people have to go back to teh drawing board on encrypting their data.
Re:This proof has already gone down in flames (Score:5, Insightful)
There must be hunderds of these "final proof of Riemann hypothesis" claims on the web. It is sad that "a Swedish newspaper is the only one to take up the story yet" doesn't inspire caution in the /. editors but urges them on to more recklessness.
Re:Breaking Encryption? (Score:3, Insightful)
Re:Don't get too excited yet... (Score:2, Insightful)
Absolute bollocks. It will be only accepted as valid when it accepted as valid by the editors of a relevant peer reviewed journal, where the reviewers are experts in the field.
Any old fuckwit can post nonsense to ArXiv. That doesn't mean a year later it suddenly becomes valid or accepted.
What peer-reviewed journal do you claim this has been submitted to? I can't see any reference to anything apart from ArXiv, which (a) isn't a journal and (b) isn't peer reviewed.
YAW.
Re:um... (Score:4, Insightful)
Please, I get enough of that kind of attitude on TV,
in movies,
at work,
at home,
at church,
at the bus stop,
in stores,
at the DMV,
in restaurants,
under my bed
and standing next to ANYONE who has vowels in their names.
Thank you.
Re:Ok for the laymans (Score:2, Insightful)
So far you're not too far off base
Right now public encryption works because it would take so long to break the encryption, even with say... 300k computers (distrubted.net) that when the encryption was broken (5-10 years later) the info would be old and it would not matter.
OK, you show that you possess basic logic skills
If someone could figure out some way to factor primes (which is along the lines of riemans sums) then they could possibly break even our stronger prime based encryption, which would make a great many people have to go back to teh drawing board on encrypting their data.
Here you make a grevious error, factoring primes is NOT "along the lines of Riemann sums." Riemann's Hypothesis, which relies on Riemann's zeta function, has little or nothing in common with Riemann Sums beyond...Riemann. Your high school calculus course did NOT teach you how to prove Riemann's Hypothesis. As a matter of fact, it's due to people like you that headlines such as "Great theorem X is proven by 26 year old construction worker with minimal understanding of multi-variable calculus," keep appearing and being proven to be WRONG. Please do a little background reading before you present your analysis to "the laymans." As a side note, I do have a degree in mathematics, though admittedly only a B.S. and even with that background, I don't feel comfortable enough with the proof to present myself as an authoritative source. However, I do know enough to point out that your statements are incorrect.
This article and the flood of uninformed and ignorant responses bring to light the need for a moderation system that provides mod points to people who actually understand the article.
Will they get the prize? (Score:3, Insightful)
And, what if the standard of refutation? Is it enough to claim "oh, this proof is all just handwaving", or "this proof is worthless, it uses a physicists approach", or does any detractor need to precisely pinpoint where the error is "on page 13, where they get from equation 63 to 64, they effectively multiply both sides with zero"?
Indeed, it appears that most mathematicians don't take the proof seriously, which also means that nobody is taking the time to check it through... Thus ironically, our hoaxters may be able to collect... which will turn out very embarrassing to the contest board, if 5 years from now some bored math student goes through it line by line, trying out all possible interpretations, and does find the error(s)...
Re:What are the security concerns? (Score:3, Insightful)
You could, for instance, adopt the following strategy: assume that the conjecture is true. Use it (however it supposedly makes encryption easier to crack) to crack encryption algorithms. If it works, great. If not, you can still crack it the old-fashioned way.
As I understand it, we're already pretty damned convinced that this conjecture is true, and we're just lacking a 100% rigorous proof of it. I don't see how the presence or absence of a 100% rigorous proof will have any effect on whether or not it's useful in cryptanalysis. Even if the conjecture turns out to be false in general, it is known to hold for an absolutely enormous set of numbers, right? Even if it only works on a small percentage of cases, that's still a small percentage of cases you don't have to solve via brute-force.
This fact tends to make me believe that the conjecture will not, in fact, help us crack encryption faster -- because we would already be using it if it helped. Could someone with a real mathematical background explain how a rigorous proof of the conjecture would make any difference whatsoever in cryptanalysis? How exactly does it apply, and how would a rigorous proof make it apply in ways that it doesn't today?
Re:um... (Score:5, Insightful)
So if it could be used to break encryption keys as you say, we would not need a proof to start doing so. We could just use it now to generate keys; and its effectiveness would be evident. So no, I don't think that the proof of Riemann's Hypothesis has any sort of bearing on encryption algorithms.
Now it could be that some of the techniques used in the proof itself could provide some insights into prime factorization methods. But again, we don't need a proof itself to get those insights, we just need the techniques themselves.
Re:No one noticed this? (Score:1, Insightful)
Besides, everyone knows the problem is undecidable anyway :)
Re:um... (Score:2, Insightful)
Don't get your math from the Cryptonomicon, get a textbook.
Re:HINT: Go read the comments on the previous arti (Score:2, Insightful)
Second, for something to be appreciated in a meaninful way, one needs to have some background on the matter.
With many scientific ideas, what happens is that the underlaying idea is relatively simple and capable of being described trough analogies, specially in graphical terms.
In some cases the dificulty lies in giving an formal definition of something relatively obvious. That's the case of manifolds: one can think about a sheet, and one will have a very close idea of the mathematical object being defined as "a locally euclidian Hausdorff topological space of dimension 2 ..."
.
On the other hand, there are many concepts (at least in mathematics) whose motivation lies in very subtle considerations pertaning highly abstract objects, without any resemblance of some day-to-day phenomenon, and those are the most hard to explain to a layman.
He proved his own point (Score:2, Insightful)
You seem to imply that the parent post proved your point by in fact explaining to you in layman term what it is that he is doing. He hasn't, read his reply again. He hasn't proved either that he does not understand what he is doing, so the comment of your first paragraph is incorrect.
It is quite possible to work in a field, make contributions and still not `understand' the field perfectly in the sense that you mean (being able to explain it to lay people) - people working in quantum mechanics would have very great difficulty in explaining to anybody what an electron *really* is, because no one knows for sure.
Einstein famously remarked that he `understood' his own stuff on general relativity in "rare moments of exceptional clarity" yet no one disputes he didn't know what he was talking about.
All this to just say that your criteria about what constitues `truly knowing' is a little bit naive IMHO.
Re:Will they get the prize? (Score:3, Insightful)
There is a saying sometimes employed by cruel mathematicians to describe illucid 'proofs': "This isn't right. This isn't even wrong."
Having said that, it surely would be nice to see 'exactly where it goes wrong'. For simple arguments a standard "the first error is on page t, line s" will suffice. More convenient is to find a claim to which an explicit counterexample can be constructed, that is, to show that the proof (if valid) leads to a contradiction - of course this works best when the proof is not reductio ad absurdum
Finally, we should always remember that the burden of "proof" is on the "prover" - it's not anyone else's responsibility if the paper isn't even written in what we'd call mathematics. (Naturally it has no chance of being published in a print journal if it's in that state, which is a necessary condition for the awarding of the prize.)
(The rest is a slightly tangential discussion of two common problems arising from extremely imprecise methods, aka "handwaving".)
Here's a gratuitous example:
Prop. There are more real numbers than integers.
"Proof." Consider the interval [n,n+1) for arbitrary integer n. In this interval there is one integer, but a slew of reals, eg, n+1/2, n+pi/4, and so forth. So, there are more reals than integers.
Of course this proposition is true but the proof is nonsense, since we can "derive" by mimickery
Prop. There are more rational numbers than integers.
which is just false.
This is a pretty simplistic example and doubtless the author of the purported proof of RH is using much more sophisticated handwaving, for which I'll produce another analogy:
Prop. 2 has no proper factorisation.
"Proof." The only integers dividing 2 are 1, -1, 2, -2, the former two being units and the latter two the number and an associate.
This proposition falls into the "not even wrong" category. Why? -- because to write down the word "factorisation" begs the question: in what ring (ie, algebraic context)? Absolutely, 2 is irreducible in the integers. But not in the Gaussian integers:
Prop. 2 is not irreducible in the Gaussian integers.
Proof. We see easily that 2 = (1+i)(1-i). Each factor of the RHS has norm 2, so neither is a unit, and in fact 1+i is a Gaussian prime, and 1-i = (-i)(1+i) is also a Gaussian prime.
The error in the first proposition about 2 was that it made no reference to the context of the discussion. This is the sort of handwaving that occurs in the "proof" of RH. We can see it really makes a difference - in one context the prop. was true, and in another false; sometimes even to utter the word "factorisation" is to already condemn oneself to the "not even wrong" bin.
Re:Will they get the prize? (Score:3, Insightful)
This proposition falls into the "not even wrong" category. Why? -- because to write down the word "factorisation" begs the question: in what ring (ie, algebraic context)? Absolutely, 2 is irreducible in the integers. But not in the Gaussian integers:
Context is everything. And in the absence of meaningful context, assume the most simple meaning of factorisation is intended, i.e. decomposition into plain vanilla positive real integers.
Not specifying context is by itself not necessarily an error. It's just sloppy writing, and makes the proof harder to read (the reader has to figure out what exactly is meant). Not specifying context only becomes an error if you start mixing to contradictory meanings. For example, if in your "factorization of 2 problem", you start two lines of reasoning, one in which you limit yourself to integers, and one in which you allow complex numbers.
A smart reader (and who also has lots of time on his hands...) may check out the proof by trying out the various possible contexts. Either he finds one context where the whole proof makes sense, or he does indeed find an incorrect mixing ("on page 10, the authors work in the algebraic context of simple integers, while on page 15 they work in the context of Gaussian integers"). Of course, the problem here is that the thing is so sloppily written, with so many underspecified contexts that nobody is really willing to invest any time debugging it...