Slashdot Log In
More on Riemann Hypothesis
Posted by
michael
on Tue Jul 02, 2002 09:35 AM
from the math-geeks-rejoice dept.
from the math-geeks-rejoice dept.
Anonymous Coward writes "The NYTimes has a little story on a recent conference at New York University's Courant Institute where mathematicians gathered to discuss potential attacks on the Riemann hypothesis. The Clay Mathematics Institute had announced an award of a million dollars for a proof (or refutation) of the Riemann hypothesis during the millenial celebrations. That million dollars won't be worth much if it takes as long as that Last Theorem by Fermat to solve. There were some interesting observations such as the statistical distribution of the zeros looked just like calculations on the energy levels of large atoms." We did a related story on hard math problems two years ago.
This discussion has been archived.
No new comments can be posted.
The Fine Print: The following comments are owned by whoever posted them. We are not responsible for them in any way.
Not more attacks! (Score:4, Funny)
You can't stop these attacks (Score:5, Funny)
Even if you are able to get into a cell it can be extremely difficult to stay in and keep your sanity. Many people who do get in just sort of drift off from society and are all but lost. Those few that make it often end up working alone, late at night in the back of dimly lit coffee houses.
There is simply no way to stop someone who is willing to make such sacrifices.
Eureka! (Score:4, Funny)
Re:Eureka! (Score:4, Funny)
Forget bigger numbers, how about smaller words? (Score:4, Interesting)
My understanding of the article is that:
A) You can't predict prime numbers.
B) That guy predicted prime numbers.
C) Alot of money goes to whoever proves how the hell he predicted prime numbers.
Ca)If we know how he predicted them we can crack old codes and make new ones?
Re:Here it is in small words (Score:5, Informative)
Re:Here it is in small words (Score:5, Informative)
The function had already been discussed by Euler.
For some reason, primes always plot along one of the axes. No one can figure out why.
Actually, that's easy. Primes (at least over the integers) are real numbers, and the zeta function maps real numbers greater than one to real numbers, which is evident from the definition as a Dirichlet series.
Quite a few proofs in analytical number theory rely on the fact that in certain areas on the right side of the line {z : Re z = 1/2} contain no zeroes of the zeta function. So far, mathematicians have tried to carefully choose these areas in order to get good results (so that you can still use them efficiently, but you can also prove that no zeroes lie in it). If we knew that no such zeroes exist at all (the Riemann Hypothesis), we could avoid all these rather technical details and theorems would improve considerably as well (for example, the error term in the Prime Number Theorem).
Re:Could you get a bit more arrogant please? (Score:5, Informative)
I'm about halfway through writing up my PhD thesis on some applications of homological algebra to knot theory and low-dimensional geometric topology (provisional title liber rerum dementiae, but it'll probably end up being called something more mathematically appropriate).
In principle, yes, I could explain the details of my research to a suitably motivated layman. But I suspect it would take rather a long time.
You see, and this really isn't meant to sound arrogant, supercilious, or dismissive, university-level mathematics is pretty damned difficult, and the details of most cutting-edge research really doesn't make sense until you've spent several years learning the background (the mindset, the language, the fundamental concepts).
My current area of research is essentially the applications of homological algebra to knot theory and low-dimensional geometric topology. To explain this to a non-mathematician, I'd first have to teach them a lot of background stuff (group theory, a bit of stuff about rings and modules, point set topology, basic algebraic topology (the fundamental group, (co)homology theory), some geometric topology (basic course in knot theory, some stuff about 3-manifolds), a bit of category theory, and some homological algebra (broad overview of the (co)homology theory of groups and algebras)).
It's taken me nearly nine years (3-year BA, 1-year MSc specialising in topology and knot theory, plus nearly five years doing a (part-time) PhD) to get to this point myself. If I were a bit cleverer (or didn't have a `proper' job as well) I might have been able to shave a couple of years off that.
My friend Steve has a physics degree. I managed, in ten minutes one evening, with much handwaving, to give him some idea of what my thesis is all about. It helped that he knew what a group was already though. But for me to explain it fully to him would probably necessitate him doing at least one mathematics degree first. And that's not really something I'd wish on one of my friends
Now this really isn't meant in an arrogant way, and I hope you won't read it like that, but Euclid was right: There is no royal road to geometry.
I can have a go at explaining the Riemann hypothesis, though. To fully understand what it's about and why it's so damned difficult you'll need to do an advanced course in complex analysis (which isn't my field either).
A complex number is a sort of two-dimensional number, which you can regard as a point in a plane (the `complex plane' or `argand diagram'). You add them together coordinate-wise, and you multiply them together in a weird manner which involves something which behaves like a `square root of -1' (engineers also like to think of it as a sort of 90-degree phase-shift operator, I'm told).
There's a particular function (`Riemann's zeta function') defined on the complex plane (it takes one complex number as input and returns one complex number). For some complex numbers (`the zeros of the function'), the value of this function is zero.
The `trivial' zeros occur at the points -2, -4, -6,
The `non-trivial' zeros (that is, all the other points for which zeta is zero) all seem to occur on the line parallel to the vertical axis that intersects the horizontal axis at +0.5. Indeed, nobody's ever found one which doesn't.
The Riemann Hypothesis is that *all* the non-trivial zeros lie on this line. It's known to be true for the first (large number which temporarily escapes me), but it turns out to be phenomenally difficult to prove that it's true in every case.
Now that's the basic idea, but it doesn't (and I can't - it's not my field) explain *why* it's so difficult that some of the greatest minds (Hardy, Littlewood, Ramanujan, etc) of the past 150 years have failed to prove it, and why the Clay institute are willing to pay a million dollars to someone who can.
- nicholas (we don't just sit around doing big sums, you know
Re:Forget bigger numbers, how about smaller words? (Score:4, Informative)
B) That guy predicted prime numbers
Riemann discovered a function that reasonably well matches the number of primes found within long intervals of numbers. It can't find primes per se, it just predicts how many you'll find between 'm' and 'n'. And it's no help for factoring a product of two primes, so it won't crack codes.
Of course, winning the prize (by taking Riemann's work a few steps further) might happen to suggest a method for factoring a product of primes, but it's more likely it will be of interest only to those few mathematicians that can remember what Riemann's hypothesis was in the first place. (I used to know but no longer remember, and that d!@#d article didn't give an equation or otherwise say anything really useful.)
Rainman Hypothesis? (Score:2)
Log in blues? (Score:1, Informative)
User
Password
Re:Log in blues? (Score:4, Interesting)
I keep a spreadsheet of user info I give when I register with various sites. All the data is serialized in a way which I can instantly identify which source the data came from. Its pretty effective for tracking down spam.
I've never had any spam directly from NYT, but they have clearly sold the information to 3rd parties, and those 3rd parties have probably sold it again and again.
I used two methods to verify it came from NYT - the e-mail address was a numbered account made on my own mail server, and my last named had a serial number appended. Both elements are present in the same I've traced to them.
The spam has come from:
- company trying to sell me HGH
- company trying to sell me diploma's
- company trying to sell me financial services
So just because they don't spam you doesn't mean that haven't enabled someone else to do it.
Proofs delicate? (Score:2, Interesting)
Wha-wha? I was under the impression that proofs are rock-solid demonstrations of a particular fact given a set of well-defined mathematical laws . . .
Re:Proofs delicate? (Score:4, Insightful)
Think about it in terms of spacecraft. A couple of vehicles were perfect and landed on Mars. One had a small defect, it wasn't complete (meters and miles were mixed up). It was lost.
Perhaps too far... (Score:1)
I know various people complain when Slashdot re-posts stories, and it's good to see you're taking note and warning us, but that's going a little too far back.
Maran
Why so much money? (Score:1)
But it's easy to prove... (Score:3, Funny)
Full Circle (Score:2, Interesting)
Someone here is clueless, but whom? (Score:2, Informative)
There is no simple way to tell if a number is prime, and that is the basis for most modern encryption schemes. Solving the hypothesis could lead to new encryption schemes and possibly provide tools that would make existing schemes, which depend on the properties of prime numbers, more vulnerable.
AFAIK, modern PK encryption depends on either the RSAP (RSA problem), related to the IFP (integer factorization problem), or the DHP (Diffie-Hellman problem), related to the DLP (discrete logarithm problem). (Then there are elliptic curves, but those aren't used much except in some proprietary systems; they haven't been studied as much and therefore aren't considered as trustworthy.)
I fail to see how breakthroughs in prime distribution theory would affect either the IFP or the DLP, or lead to new cryptosystems. So, am I clueless, or has the NYT done it again?
It's the Times... (Score:5, Informative)
...and your description of the mathematical basis for modern encryption is essentially correct. One could argue that there is a relationship between finding out if a number is prime and determining its prime factors. But such a relationship has so far eluded mathematicians.
The statement "there is no simple way to tell if a number is prime" is true only for a limited definition of "simple way." The impossibly complex ways of the past have been replaced with complex-but-definitely-possible techniques, which would definitely make encryption vulnerable if encryption depended on the properties of prime numbers. Which it doesn't. (As you correctly point out.)
Another quote from the article (paraphrased by the submitter), is also erroneous unless something has changed:
I believe the Clay Mathematics Institute award specifically excludes refutation. (Presumably because someone could refute the hypothesis simply by stumbling onto a single counter-example: a zeta-function zero which does not lie near the complex axis. This would be the mathematical equivalent to hitting the lottery and might do little to advance mathematics.)
Prove this (revised) (Score:1, Offtopic)
Step One: Steal Underpants
Step Two:
Step Three: Profit.
When they can prove that, THEN I'll be impressed.
When I finsh my Linux Xbox port (Score:2, Funny)
A proof that is worth millions to MAN kind (Score:2, Funny)
Keep in mind this proof looks much better if you can actually use the square root symbol
The problem:
Prove that women are all evil.
(With written proof, men don't have to worry about women arguing this fact anymore
The proof:
Given that:
Proceede with the proof:
See what an undergrad in Mathematics, an undergrad in C.S., and a Master's in C.S. gets you
Seriously, I wish someone could prove that P=NP. I hated graduate Algorithms! This would have eliminated a portion of my least favorite topic in that course (NP and NP-completeness). If this world is not truely hell, someone will prove that and share it to help prevent the suffering of innocent C.S. graduate students.
Let's not be too hasty (Score:1)
Re:Let's not be too hasty (Score:4, Informative)
Secondly, iirc, Gödel showed that sufficiently complex systems have to either be inconsistant or incomplete using a very specific paradox
Finally, who's being "hasty"? What exactly are you suggesting? That they give up the search for a proof because there's a tiny chance that it may be unprovable? Why doesn't the entire field of theoretical math just stop right now, then?
ZetaGrid (Score:5, Informative)
Apparently there's a distributed computing project called ZetaGrid [hipilib.de] which has calculated the first 50 billion zeros out ... if you're bored of SETI@Home, this might be a nice change of pace.
Riemann Hypothesis [wolfram.com]
Riemann Zeta Function [wolfram.com]
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.
I am confident... (Score:3, Funny)
Either that, or you can solve them by buying REAL ESTATE with NO MONEY DOWN! or by placing SMALL ADS in NEWSPAPERS with your own 900 NUMBER!!!!!
Good intro... (Score:5, Funny)
That has to be the funniest things I've read, today.
Is it me or does it seem that all "hard" mathematicians are either at war with God or trying to "refute"/"prove"/divide/discover/humiliate him/her/it/Taco?
Here's some background info... (Score:2, Redundant)
Riemann Hypothesis [wolfram.com]
karma whore (Score:1)
Amazing new result from geometry (Score:1)
Researchers at a leading US university have made an astounding discovery. They have found that the square length of the hypotenuse of a right angled triangle triangle can be found by adding the squares of the hypotenuses of the other two sides.
Dr. P Thagoras explains: "we've experimented with many kinds of right angled triangle it it seems to hold in all situations." Prof. E Clid is enthusiastic about the applications "for example a builder can predict the length of the diagonal of a plot of land withput actually measuring it. We can run the software to compute it from the sides on something as small as a laptop. A builder could easily have one of these on the actual building site."
Of course the discovery is not without skeptics. "They haven't tested every triangle", says Dr. P Appus, professor of post-modern sociology, a researcher who studies scientists themselves. "These researchers have only picked those triangles that fir the pattern. It's a kind of unconscious Freudian repression where triangles that don't fit are collaboratively eliminated from the field of view in a reactionary social construct".
But Thagoras isn't disheartened. He believes gis result might hold even for really big triangles. "I think you could use this when urban planning. I bet it'd hold for triangles miles across".
A bold claim, and only time will tell whether these claims will hold. But don't expect to see builder wielding those laptops any time soon!
Those Damn terrorists... (Score:1, Funny)
harmony (Score:3, Interesting)
Of course primes have a generally log distribution, because every prime you find provides a factor later on down the line so the primes become more sparse.
Then there's the atoms thing, sfaik shells/energy levels are basically harmonic and a harmonic is more-or-less the opposite of a prime.
since harmonics and the increasing sparseness of primes could be taken as identical you're going to get the same distribution patterns out.
here goes
primes v harmonics
2 is prime and a harmonic root
3 is prime and a discord (root)
4 is non prime, and the second octave of the first root
5 is prime and a discord (root)
6 is non prime, and cord of the first and second roots
7 is prime and a discord (root)
8 is non prime, and third octive of the first root
9 is non prime, and first octave of the second root
etc....
ANKOS to the rescue! (Score:2, Interesting)
Value of a million dollars. (Score:1)
Okie doke, forgive me if I'm missing something here, but is Fermat's Last Theorem the same as the conjecture mentioned in the article? The one that took Dr. Andrew Wiles seven years to solve? If so, why would $1 million not be worth much in 7 years?
There's two ways to look at this. The first is, how much money do I expect to make in the next seven years? I calculated mine, assuming I continue to get the same percent pay raise for each of the next seven years, and let's just say, I won't have made my first million for a few years after that unless we get another dot-com boom or some other such aberation.
The other way is, how much will a million dollars in today's money be worth seven years from now? Assuming the inflation rate for the next seven years matches that of the previous seven years, it'll be worth approximately $850,000 (see this inflation calculator [bls.gov]).
So, why will $1 million dollars be a paltry sum in seven years?
Regarding the Clay Math Institute "Business Model" (Score:1)
In particular, is Landon Clay free to spend some of the interest on the millions he has supposedly "donated" to math through these prizes in any way that he pleases, so long as a fraction of the interest is spent on some tenuous connection to "promoting mathematics". (Check out the link on the CMI webpage to the Clay-sponsored yacht cruise in the Boston Harbor.)
Rumor has it that the president of the Clay Math Institute was fired by the Harvard Math Department for spending too much time shaking Clay down for umpteen millions, and not enough time doing research. Can anyone provide a confirmation of this rumor? Furthermore, after being "dismissed" from the Harvard Mathematics Department, the president of CMI mysteriously popped up across the river at Boston University. Does anyone know how much the Clay Math Institute has donated to Boston University in the process?
Finally, more to the point of the Riemann Hypothesis, which we all want to see solved, what are folks' opinions about whether a $1M prize on the problem is likely do more to decrease the likelihood that a solution is found sooner than later, given that the money will create less incentives for researchers to share their insights and conferences or publish partial results in journals?
Personally I think the prizes smell too much of Clay's past career in the actively-managed mutual fund business, where it's all about out-performing the index for that bonus at the end of the year. Perhaps the first bit of math that Clay should learn (he supposedly dropped out of Harvard himself and never learned anything beyond high-school algebra) is a little statistics, which would show how an active manager's "ability" to beat the index has more to do with luck than business acumen. (Read the famous book A Random Walk Down Wall Street, or check out the site www.indexfunds.com). Then maybe he might realize the right place to "donate" his money is in the form of a refund to investors who got jipped by the front-end 5% loads they paid supposedly for Clay's investment genius. Clay's fund specializes in tax-managed investments, so I guess we can be sure that those skills for dodging the watchful eye of the IRS sure came in handy when setting up his retirement tax shelter
Anyway, I'm starting to ramble now
Re:Who stands a better chance? (Score:2, Insightful)
specifically, i'd place my bets on the smelliest and most Russian of them.
Re:Who stands a better chance? (Score:4, Interesting)
The mathematician stands a better chance of proving the hypothesis, but the NSA supercomputer stands a better chance of refuting the hyposthesis.
With current technology, it's extremely unlikely that the mathematician would refute the hypothesis or the computer might prove it (although it is possible).
Finally, props goes out to the Courant Institute of Mathematical Sciences. The best, my favorite, and my current graduate school (@ nyu).
Re:Wrong forum (Score:1)
-Ed
docbrown.net [docbrown.net]
Graphic Design, Web Design, Role-Playing Games...all the good stuff
Re:Prove This (Score:1, Offtopic)
Re:If we're so smart... (Score:1)
Oh, and Russians are not often credited blah blah any more than other large countries. The communist system often held back research because Russian scientists could not access Western research, and they often couldn't publish either. That's no way to do science.
The point with this is that the Riemann hypothesis shows that the distribution of prime numbers shows an underlying pattern which defy current understanding. The connection with quantum states appears to indicate that those states are ruled by primes in some unknown way. It's a bleeding complex area of maths that few of us here (damn, few people full stop) have a good understanding of. Merely repeating the dumbass "If we're so smart, how come we don't know this yet" gets us nowhere.
Re:If we're so smart... (Score:1)
So although I don't know if mathematicians today are better or worse, I don't think you can draw a conclusion just because they haven't been able to solve an old problem. It is often easier to pose a difficult problem then it is to solve them.
Re:Sounds like a job for!.... (Score:1)