# Two breakthrough results in number theory

During the past two weeks, two truly major results were announced in the realm of (analytic) number theory, namely the mathematics of integers in general and of prime numbers in particular. Prime numbers, i.e., 2, 3, 5, 7, 11, 13, 17, 19, 23  … are the building blocks of arithmetic and have been studied seriously since before the time of Euclid (c. 300 BCE), when it was already proven that the primes were infinite in number.

Problems about prime numbers are fundamental and often intractible. In his article “The first 50 million prime numbers” in the Mathematical Intelligencer (1977), Don  Zagier wrote

…there is no apparent reason why one number is prime and another not. To the contrary, upon looking at these numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation.

So real progress on the understanding of prime numbers  is something to celebrate.

### Twin prime conjecture

The first of the two latest results concerns what is known as the twin prime conjecture, namely that primes just two apart, such as 3 and 5, 11 and 13, 17 and 19, 41 and 43, etc., continue to appear indefinitely. Numerical searches appear to confirm the conjecture and mathematicians generally believe that it is true. But rigorously establishing it has proven to be enormously difficult.

There is no record of who first proposed the twin prime conjecture — it has been in the mathematical literature for centuries. Several generalizations have been proposed. The Hardy-Littlewood conjecture, for instance, says that the number of primes p for which p+2 is also prime is not only infinite, but indeed the fraction of such primes in the first n integers tends to C2 n / (log n)2, where C2 is approximately 0.6601618… See the Wikipedia article on the twin prime conjecture and an article by Sadegh Nazardonyavi for additional background.

Technical details are presented in G.H. Hardy and J.E. Littlewood, “Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes,” Acta Math. 44 (1923), 1-70, and E. Bombieri, J.B. Friedlander, and H. Iwaniec, “Primes in Arithmetic Progression to Large Moduli,” Acta Math. 156 (1986), 203-251.

On 13 May 2013, Yitang (Tom) Zhang of the University of New Hampshire, Durham, announced a proof that there are infinitely many prime numbers separated by less than 70,000,000. Now 70,000,000 is not two, but this is the first result that has established any finite bound at all. Zhang’s work builds on results of numerous earlier researchers, including Bombieri, Vinogradov, Elliot, Halberstam, Goldston, Pintz, Yildirim and others. Zhang constructed a “sieve,” not unlike a gold-panning system that processes lots of of liquid-suspended ore, leaving a few nuggets, which he then analyzed using  various advanced techniques.

Other leading mathematicians have praised Zhang’s work. Daniel Goldston of San Jose State University, who himself has worked on this problem, said that the result is “astounding.” Andrew Granville of the University of Montreal in Canada, one of the leading living analytic number theorists, declared that

This is one of the great results in the history of analytic number theory. … It’s just extraordinary. I would never have expected it in my lifetime.

He notes that it is “virtually unheard of” for such a significant result to come from a mid-career mathematician who had not previously been noted for research in this highly technical area. Many observers feel that the bound of 70,000,000 will quickly be reduced, once researchers in the field fully understand and further refine Zhang’s techniques. But while it is unlikely to lead to a resolution of the full conjecture, Granville notes

This work is a game changer, and sometimes after a new proof, what had previously appeared to be much harder turns out to be just a tiny extension. … For now, we need to study the paper and see what’s what.

Some additional details on Zhang’s work can be found in a detailed  Simons Foundation news article by Erica Klarreich, which has already been reprinted in Wired, and in a Science article by Dana Mackenzie.

[Added 24 April 2014: In the weeks following the announcement of the twin primes result, a team of mathematicians working as part of the Polymath project reduced the limit from 70,000,000 first to a few million, and then to a few thousand, and, finally, to 246. Terence Tao now says that they have reached a point of diminishing returns, so the 246 figure will probably remain for a while. See this New Scientist article for details.]

### Goldbach conjecture

The second result concerns the Goldbach conjecture, which in its strong form is that every even number greater than two is the sum of two primes, and in its weak form that that every odd number greater than five is the sum of three primes. Note, for instance that 13=3+5+5 and 36=17+19. Again, these conjectures have been studied in great detail, both mathematically and numerically, and are generally thought to be true, but there has been no proof of either. Christian Goldbach (1690-1764) was a German mathematician and lawyer known for his work in number theory and analysis. He stated the conjecture known by his name in a now-famous 1742 letter to Euler.

Various noteworthy mathematicians have worked on this problem. Most progress relies on the so-called Hardy-Littlewood-Vinogradov circle method, which in turn involves Fourier analysis, the familiar tool used by physicists and electrical engineers to decompose a periodic signal into its spectrum  (the frequencies of its harmonic components).

In the 1930s, Dhudakov, van der Corput and Estermann employed an approach of exponential sums due to Vinogradov to show that almost all even numbers can be written as the sum of two primes (in particular, that the fraction of even numbers which can be written in this way approaches unity). In 1975, Montgomery and Vaughan showed that the set of even integers which are not the sum of two primes has limiting density zero. See the Wikipedia article on Goldbach’s conjecture and the above-mentioned article by Sadegh Nazardonyavi for additional background.

On the same day as the twin prime announcement, Harald Helfgott, a 35-year-old mathematician at the Ecole Normale Superieure in Paris, announced a proof of the weak Goldbach conjecture.

In 1923, Hardy and Littlewood showed that by constructing a function whose spectrum consists of prime numbers, and then cubing it, one could establish the second Goldbach conjecture by showing that no odd-numbered frequencies are missing. Previous mathematicians had established that all frequencies above 101300 were present. Inspired by the so-called GPY paper, Helfgott was able to reduce this bound to “only” 1030.

Such a large number was once considered too big to be of any practical use, but, given modern computer technology, that is no longer the case. Indeed, Helgoft’s colleague David Platt numerically verified the required condition for every number below this limit, a computational tour-de-force that required 40,000 CPU-hours of computer run time (about 4.5 CPU-years). To put this in context, the present bloggers recently expended nearly 2000 CPU-years of BlueGene computation time on an interesting but much less worthy cause, namely computing inaccessible bits of certain constants.

Fields medalist Terence Tao of the University of California, Los Angeles, who last year proved that any odd integer is the sum of at most five primes, cautiously endorses the result, although, as with the twin prime result mentioned above, it must survive the scrutiny of careful review by several leading mathematicians.

Some additional details on Helfgott’s work can be found in the above-mentioned Science article, and also in a blog post by Artem Kaznatcheev and Kate Zen. Helfgott’s latest manuscripts are available here and here.

### Some concluding observations

Analytic number theory has seen much productive usage of high-powered computation. In the case of the Green-Tao Theorem on the existence of arbitrarily long arithmetic sequences of primes, it was the computation of a sequence of length 24 that convinced the authors the result was true.

Additionally, it was computation regarding the twin prime distribution that in 1994 unveiled the Pentium bug. That said, any number-theoretic conjecture that relies on very slowly growing functions and even vast ammounts of seemingly good evidence can be misleading or unconvincing. Such is the case for much of the computational work on the Riemann hypothesis.

It is also worth noting the analogy of the Goldbach proof to that of the Four Color Theorem, namely the assertion that, under certain reasonable conditions, only four colors suffice to color any planar map so that no two adjacent regions have the same color. The only proofs of this result to date have required computer-assisted analysis of many individual cases. Will Helfgott’s result or the Four Color Theorem eventually be proven without the aid of computation? Perhaps. But, as we argued in the first chapter of Mathematics by Experiment, there is no guarantee that such a proof exists.

What is also interesting about both of these landmark results is that neither involves a big new idea. Rather, they rely on painstaking, careful and incremental application of previous work (the shoulders of giants).  As was noted in the Science article, a Prime Week for Number Theory,

[a]fter generations of glacial progress on both problems, however, number theorists like Roger Heath-Brown of the University of Oxford in the United Kingdom were delighted to see signs of a spring thaw. “Despite hundreds of years of history, number theory is still moving on,” Heath-Brown says. “That’s what I find most exciting about these two theorems.”

As such they are more representative of how most mathematical progress takes place; not, as is commonly believed, from earthshaking, paradigm shifting ideas by very young, primarily male researchers.