You wait ages for number theory results, then two come at once

We should celebrate recent progress on the understanding of prime numbers. Pawel Bak

In recent weeks, two truly major results have been announced in the realm of (analytic) number theory, namely the mathematics of integers in general and of prime numbers in particular.

Number theory, as you no doubt know, deals with the properties and relationships of integers, or “whole” numbers.

Prime numbers – those that can be divided evenly only by 1 or themselves – are the building blocks of arithmetic and have been studied seriously since before the time of the ancient Greek mathematician Euclid, when it was already proven that the primes were infinite in number.

Problems relating to prime numbers are fundamental and often easy to state but intractable. In his 1977 article The First 50 Million Prime Numbers, the American mathematician 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 recent substantial 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, which posits that primes just two apart – 3 and 5, 11 and 13, 17 and 19, 41 and 43, and so on – continue to appear indefinitely. Numerical searches appear to confirm the conjecture and mathematicians generally believe that it is true.

But on May 13, 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.

Clearly, 70,000,000 is not two, but Zhang’s is the first result to establish any finite bound at all. The video below is instructive for those who wish to know more about the result and its significance:

Gaps between Primes.

Many leading mathematicians have praised Zhang’s work. Daniel Goldston of San Jose State University, who has also worked on this problem, described the result as “astounding.” Andrew Granville of the University of Montreal, one of the leading living analytic number theorists, declared described it as:

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.

Granville also noted that it was “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.

Goldbach conjecture

The second result concerns what’s known as the 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 every odd number greater than five is the sum of three primes. Note, for instance, that 13=3+5+5 and 36=17+19.

As with the twin-prime scenario, these conjectures have been studied in great detail, both mathematically and numerically, and are generally thought to be true, but there had been no proof of either.

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.

Previous mathematicians had established that all frequencies above 101300 were present. Inspired by the so-called GPY paper – a paper by the American mathematician Daniel Goldston and two colleagues which introduced “a method for showing that there exist prime numbers which are very close together” – 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’s 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).

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

An overview of the Goldbach conjecture, including some information on Helfgott’s recent contribution, can be gleaned from the video below:

The Goldbach conjecture.

Putting the news in context

Interestingly, neither of these landmark results involves a big new idea. Rather, they rely on painstaking, careful and incremental application of previous work – on the shoulders of giants. As was noted in a recent Science article:

[a]fter generations of glacial progress on both problems […] number theorists […] were delighted to see signs of a spring thaw.

As such, the new results are representative of how most mathematical progress takes place – slowly and sure-footedly, and not, as commonly believed, from earth-shaking, paradigm shifting ideas by very young, primarily male researchers.

There is hope for the rest of us!


A version of this article appeared on Math Drudge.

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