Riemann hypothesis proved ?

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

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.

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

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.

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.

Walt — so there should be infinitely many numbers x for which both x and x + 2 are prime? This seems intuitively dubious. (Which means nothing, but) — I always think of primes greater than 7 as being quite sparse.

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.

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.)