Summer Break!

Blog HQ is closing up shop for the summer.  But don’t despair we’ll be back mid-August!

From Spiked Math.

But before we go, we have to point you to two new cool results in number theory.

The famous Twin Primes Conjecture says that even though on average the gaps between successive prime numbers becomes larger and larger (indeed, you can find gaps of arbitrarily large size), there are also infinitely many pairs of prime numbers which are only two apart (like 3 and 5, or 11 and 13).  Nobody knows how to prove this, but Yitang Zhang just proved that there are infinitely many pairs of prime numbers which are less than 70,000,000 apart!  This is a major breakthrough! Of corse, seventy million is a long way from two, but since there was no bound before Dr. Zhang’s theorem, it’s a huge leap forward towards proving the Twin Prime conjecture.

To read more about Dr. Zhang and his result, read the excellent article on the Simon’s Foundation website.  There is a more detailed explanation of the proof in this blog post about a recent talk by Dr. Zhang.

You might guess such a result would be proven by a famous Field’s medalist but, remarkably, Dr. Zhang is relatively unknown among mathematicians.  As the famous number theorist Andrew Granville put it:

“Basically, no one knows him,” said Andrew Granville, a number theorist at the Université de Montréal. “Now, suddenly, he has proved one of the great results in the history of number theory.”

– From the Simons Foundation article.

The other big result (which would have been headline news if it wasn’t for Dr. Zhang!) is Harald Helfgott‘s recent proof of the “weak” Goldbach conjecture that every odd number greater than five is the sum of three primes.  We talked about the “strong” and “weak” Goldbach conjectures here. You can go there to read more about the conjecture, but suffice it to say that it is right up there with the Twin Prime Conjecture as a long standing open problem in number theory.  We were happy to discover that Dr. Helfgott is a fellow blogger!

The Goldbach Conjecture

Not to be confused with the The Gold-Bug, the Gold Bug Variations, or the Goldberg Variations, the Goldbach Conjecture is a perhaps second only to Fermat’s Last Theorem among the famous unsolved problems of mathematics.  In fact, it’s at the center of at least one book (Uncle Petros and Goldbach’s Conjecture) and one film (Fermat’s Room)

Like many questions in number theory, it’s ridiculously easy to explain to someone over cocktails, but even more ridiculously hard to solve.

Christian Goldbach

What is it?

The better known version is the “strong” Goldbach Conjecture:

Every even integer greater than 2 can be written as the sum of two primes.

The other version is the “weak” Goldbach Conjecture:

Every odd integer greater than 5 can be written as the sum of three primes.

If you think about it for a minute (and subtract 3 ) you’ll realize that if the “strong” conjecture is true, then the “weak” conjecture is also true.  That’s why they’re called “strong” and “weak”, respectively.

From Wikipedia, the history on the problem is thus:  On June 7th, 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he told Euler of his conjecture.  In reply, Euler wrote

“Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe necht demonstriren kann.” (“every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.”)

In a couple of weeks it will be the 270th anniversary of Goldbach’s Conjecture and it’s still unproven!

Not for lack of trying.

It being the modern era, people have checked it on computers.  In particular, Tomás Oliveira e Silva ran a computer search that verified the conjecture for all numbers less than 1,609,000,000,000,000,000 (P.S. He didn’t find a counterexample).

Computer calculations reveal something else:  If you make a plot where the x-axis is your number n and the y-axis is the number of ways to write n as a sum of two primes, you might expect the graph to be random looking.  However, when you plot the graph you discover that the you actually get a pretty cool looking graph called Goldbach’s comet:

Goldbach’s comet (thanks to Wikipedia)

What about on the theoretical side?  Various partial results are known (see Wikipedia for details).

In 1995 Ramaré proved that every even integer is the sum of six primes (so a weaker version of the “strong” Goldbach conjecture).  In 1995 Kaniecki proved that every odd integer is the sum of at most 5 primes (so a weaker version of the “weak” Goldbach conjecture)*.  Unfortunately, Kaniecki’s proof depends on assuming that the Riemann Hypothesis is true.  The Riemann Hypothesis is, of course, another equally famous unproven result in mathematics.

Recently, Terry Tao proved that every odd integer is the sum of at most 5 primes.  The key thing is that he doesn’t have to assume the validity of the Riemann Hypothesis to do it.  So Kaniecki’s theorem is true without any qualifications.  It’s a very nice result and Dr. Tao posted about it on his blog.**

Because it’s nice math and because Goldbach’s Conjecture can be explained to your Uncle Dave, Terry Tao’s paper has gotten some play (for example, on Scientific American and Slashdot).

We would be remiss if we didn’t point out there is an internet full of purported proofs Goldbach’s conjecture.  For example, here, here, and here.

* But notice that Kaniecki’s theorem implies Ramaré’s theorem if you subtract by 3.

** It being the Internet, the third comment on Dr. Tao’s blog is a complaint about his choice of title for the paper .