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 it would be 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 on my blog 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.

{ 18 comments }

Jason Kuznicki 06.11.04 at 4:34 am

I’m not a mathematician, but what about the Goldbach conjecture?

Michael Kremer 06.11.04 at 4:45 am

Concerning the purported proof of the twin prime conjecture, the following is from http://www.youngmath.net/concerns/story/2004/5/29/131042/934

“Update [2004-6-10 13:39:44 by overconvergent]: The preprint has been withdrawn from the arXiv; because:

A serious error has been found in the paper, specifically, Lemma 8 is incorrect.”

So that one is still unsolved too.

John Quiggin 06.11.04 at 4:49 am

I’m not a mathematician either but I don’t think the Goldbach conjecture is comparable in significance to the others I mentioned – it’s more like the twin primes problem.

Kevin Drum 06.11.04 at 4:50 am

De Branges has been claiming to have a proof of Riemann on a periodic basis for a very long time. Treat this one with a very big grain of salt for now.

Michael Kremer 06.11.04 at 4:50 am

I now see that the Mathworld article linked to in the original post also points out the likely error but is optimistic that it might be fixable. See the last paragraph of the report.

Aaron Bergman 06.11.04 at 5:34 am

The twin prime proof has been withdrawn.

Giles 06.11.04 at 5:44 am

I thought that the travelling Salesman problem is still out standing – its a mathematical problem that will also be very useful

Kieran Healy 06.11.04 at 6:06 am

The way Mathworld put it just made me laugh out loud:

bq. 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.

Andrew Boucher 06.11.04 at 6:16 am

There are frequently claims of proofs of important theorems. Usually they don’t pan out.

The Riemann Hypothesis is not the last significant mathematical conjecture. It is probably considered the most important open mathematical problem, so if it were solved, there would have to be a new one. P = NP (what one comment refered to as the Travelling Salesman problem) would probably be a candidate, although there are also many simply stated problems still open in number theory. For those who want to get wealthy, there are the seven problems set by the Clay Mathematical Institute (RH and P = NP are two of them).

As another poster has pointed out, the recent (serious) proof of the infinitude of the twin primes apparently has a problem.

Chris Bertram 06.11.04 at 7:53 am

I note that the same Mathworld page also announces Ben Green and Terence Tao’s proof of the existence of arbitrarily long progressions of primes. Green is a Bristol boy and recently gave a presentation of this in our Maths department here. My (more mathematically savvy than I) colleagues who attended came away really impressed and haven’t stopped talking about it for weeks.

Walt Pohl 06.11.04 at 8:27 am

I’m not sure the Poincare conjecture is definitely considered to be settled. Last I heard, people were still digesting the announced proof.

De Branges has a history of announcing proofs of results that later turn out to be incorrect, so who knows.

Twin primes and Goldbach are by themselves not _that_ important, but they are typical examples of whole classes of questions that we have no idea how to answer.

Let’s put the twin primes conjecture another way: Consider the polynomial x^2 + 2x. Now, value of this polynomial is never prime (for natural numbers), since it factors as x(x+2). How often is it a product of two primes? If the twin primes conjecture is true, then it’s infinitely often.

There’s nothing special about x^2 + 2x, though. We can consider any polynomial, and ask how often does it take on prime values, the product of two primes, etc. For nonlinear polynomials, we don’t know the answer. You can rule out certain cases happening by elementary algebra, but beyond that, we don’t know. If we knew the answer for any one, such as twin primes, we would probably know the answer for a bunch of others.

mat 06.11.04 at 10:05 am

The De Branges not-quite-a-proof paper was quite funny as well at the end, full of grandiloquent observations about how this proof was a

deliciousfinding because of my family motto being blablabla… etc.Jeremy Osner 06.11.04 at 10:37 am

Walt — so there should be infinitely many numbers

xfor which bothxandx + 2are prime? This seems intuitively dubious. (Which means nothing, but) — I always think of primes greater than 7 as being quite sparse.Abiola Lapite 06.11.04 at 11:52 am

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J. Ellenberg 06.11.04 at 1:48 pm

Jeremy,

No, primes are pretty common; much more common, for instance, then perfect squares or cubes, but on the other hand much less common than odd numbers. If primes were distributed “randomly” throughout the integers, one would certainly expect infinitely many twin primes. Interestingly, the Green-Tao theorem alluded to above (caveat: I haven’t read the paper so only know about the argument by hearsay) gives a precise sense to the assertion “the primes are distributed in an approximately random way” and shows that ANY sequence of integers which is “approximately random” in their sense contains arbitrarily long arithmetic progressions. In some sense, their theorem is not really a theorem about primes at all.

Anarch 06.11.04 at 3:09 pm

Jeremy’s intuition is correct in that the sum of the reciprocal twin primes converges — even if there are infinitely many — while the sum of the reciprocal primes diverges. [It’s Brun’s theorem (?), and I don’t pretend to know anything about the proof.] The catch is that “being infinite” isn’t even close to questions of sparsity or density; it’s one of the more obvious ways to concretely introduce someone to the concept that “infinity is REALLY big!”

As to this:

Iâ€™m not sure the Poincare conjecture is definitely considered to be settled. Last I heard, people were still digesting the announced proof.The proof they’re digesting, by a Russian whose name I can’t remember atm, is actually of the full Geometrization Conjecture and not merely the Poincare Conjecture. It’s entirely possible the full GC will fail but the PC will go through; again, though I’m not qualified to comment.

Steve 06.11.04 at 5:45 pm

No, primes are pretty common; much more common, for instance, then perfect squares or cubes, but on the other hand much less common than odd numbers.And, in fact, one of the side benefits of proving the Reimann Hypothesis would be that the prime number theorem could be calculated much more exactly. That’s one of the reasons it’s considered such an important question — it ties into an big unanswered question in a different branch of math. (Demonstrating that P = NP or P != NP would also be a hugely important result, but it’s less pure-math-y in my mind, and the consequences would certainly be much more directly applicable to the real world via algorithm writing.)

Walt Pohl 06.11.04 at 7:07 pm

Jeremy: I’m inclined to think the twin primes conjecture is true, but the answer is interesting whether true or false. What’s surprising is that we don’t know the answer either way.

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