According to this report, Louis De Branges claims to have proved the Riemann Hypothesis. If correct, it’s very significant – much more so than the proof of Fermat’s Last Theorem by Wiles.

It is also, I think, the last of the big and well-known unsolved problems in mathematics, and its nice to see the search ending in success. Some of the other big problems have been closed, rather than solved. The classic problems of the Greeks such as squaring the circle were shown to be insoluble in the 19th century, and the Hilbert program of formalisation was shown by Godel to be infeasible. And the four-colour problem (not a really important problem, but a big one because it was easily described, interesting and very tough) was dealt with by a brute-force computer enumeration.

**Almost instant update** Commenter Eric points to Mathworld which says “Much ado about nothing”. On the other hand, the same page reports a proof of the infinitude of twin primes which has been an open question for a long time, though not a problem in the same league as those mentioned above.

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Not qualified to comment on this, but I will quote from http://mathworld.wolfram.com/

“Riemann Hypothesis “Proof” Much Ado About Nothing

A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges’s home page seem to lack an actual proof. Furthermore, a counterexample to de Branges’s approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.”

I bought a book on the Riemann hypothesis a few months ago: “The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics” by Karl Sabbagh

Regarding Louis de Brange’s, it says this:

I could have started my researches with any of a dozen or more mathematicians who were known to be working on the Riemann Hypothesis, but I chose de Branges for the simple reason that he told me he was putting the final touches to a proof. After getting over my initial excitement, it took only two or three phone calls to other mathematicians to realize that no one else took such a statement seriously. This wasn’t the first time de Branges had put the finishing touches to a proof of the Riemann Hypothesis. There was “no chance” that de Branges had a proof, his fellow mathematicians told me; he was “always telling people he had a proof and it was always riddled with errors”; he’s “working in the wrong area” for a proof of the Riemann Hypothesis; and so on.and

“It would be easy to dismiss le Branges as a crank… but he has earned the right to a hearing because the early dismissals of his work on the Bierberbach Conjecture turned out to be wrong.” – Joe ShipmanI’m ready to take the Captain’s your word for it that the four-colour problem is not a really important one. But what is the criterion? Are teams engineers waiting for the final word on the Riemann Hypothesis before they embark on various projects that would be infeasible if it were false? Has the lack of a solution impeded progress in quantum physics or cosmology? Are there other important (by some criterion) hypotheses that rest on the Riemann hypothesis? Or will its solution just free up human capital to get on with curing cancer and so on.

Are there other important (by some criterion) hypotheses that rest on the Riemann hypothesis?

Yes. There are quite a lot of theorems that begin by assuming the Riemann hypothesis and derive various consequences from this. Of course, the link between “mathematically important” and “practically important” is rather hard to pin down. But it’s there, as all those teams of engineers doing various useful things can attest.

Why is the Riemann hypothesis important.

(1) Historically, its one of Hilbert’s 10 unoslved problems that set the agenda for 20th Century maths.

(2) theres a $US1m for anyone who proves it!

(3) A lot of other hypotheses follow from the Riemann hypothesis, most significantly, is the prime number theorem(hypothesis) which amounts to being a prediction of the occurence of prime numbers. Prime numbers have resisted all attempts at being predictable and are, consequently, the source of much mathematical angst.

But I don’t know whether it will help feed the world.

From the point of view of someone who is not a mathematician, but has an interest in mathematics, the question of what is an important theorem is an interesting one. I think that, as a crude approximation, an important theorem is one that mathematicians tell me is important.

This may be flattering to mathematicians, but I think it is reasonable. The connection between significant mathematics and practical benefit to our species is often so indirect that it is impossible to recognise before the benefit is realised. But the potential for even the most pure of mathematics to be of enormous worth eventually is indisputable. For this reason, I think it best we leave them to their research, and when they claim triumph give them the benefit of the doubt.

I recall a former lecturer of mine, frustrated at the increasing demands placed on im and his colleagues to demonstrate the worth of their research, remarking on the absurdity of beauracrats judging the worth of work they could not even understand. I think he had a point.

While it might not be quite as well known, there’s a unsolved problem in theoretical computer science with potentially enormous practical implications, particularly if we get the answer we don’t expect. The problem is the P = NP question. To an extremely rough approximation, it asks the question “if it is possible to verify an answer to a problem efficiently, is it therefore possible to find the answer efficiently in the first place?” for a very large class of problems.

There are many practical problems in this class, some of them of great interest to economists; they involve problems like production and transport scheduling. A “yes” answer would probably (depending on the exact nature of the “yes” answer) revolutionize these fields, along with many others.

Most computer scientists think the answer is, unfortunately, “no”; in other words, that for this large class of problems, it is much harder to find a solution than to check that the solution is correct. This has in fact been suspected ever since Stephen Cook asked the question back in 1971. But nobody’s been able to prove it; and nobody’s close to it (or at least has admitted to being close to doing so in public).

So, while some of the great problems of mathematics may have been solved, there are others of potentially enormous significance that still remain tantalizingly out of reach.

math is teh 1337 r0x0rz.

I think it is practically significant- the randomness of prime numbers is the basis for cryptography, and if they can be predicted programatically, this provides the basis to break encryption such as SSL- this would have a major impact on IT, intelligence, Finance etc….. a fairly fundamental significance