From some point onwards, every odd number can be written as the sum of three primes.
Here, the phrase ‘from some point onwards’ might seem to indicate that only a small number of cases remain to be checked—which is a finite task that could quickly be carried out by hand, or by computer. But in practice, the point from which the result had been proved true was massive, with millions of digits, and checking the remaining cases was way beyond the capacity of any existing computers. However, in 2013, and following much further theoretical work by several people, the Peruvian mathematician Harald Helfgott (working in France, with computational help from the British number theorist Dave Platt) managed to reduce, and eventually to eliminate, the huge gap between what had been proved for large numbers and what was already known for small numbers, giving the following result:
Every odd number that is larger than 5 can be written as the sum of three primes.
What has this to do with Goldbach’s conjecture? Well, if Goldbach’s conjecture is true, then every even number (≥ 4) is the sum of at most two primes p and q, and adding 3 tells us that every odd number (≥ 7) is the sum of at most three primes (p, q, and 3). But unfortunately the argument doesn’t work the other way around: Helfgott’s achievement doesn’t yield a proof of Goldbach’s conjecture, but it does provide a weaker form of it. For if n is an even number that is larger than 8, then can be written as the sum of three primes. Also, , and so
Every even number that is larger than 6 can be written as the sum of four primes.The twin prime conjecture
The second conjecture concerns twin primes which, as you saw in Chapter 1, are pairs of primes that differ by 2. Those up to 100 are
There are thirty-five pairs of twin primes up to one thousand, over eight thousand pairs up to one million, and over three million pairs up to one billion. The largest known pair has over fifty thousand digits!
In 1846 the following conjecture was formulated by the French number theorist Alphonse de Polignac:
Twin prime conjecture: There are infinitely many pairs of twin primes.
For many years, number theorists have tried to prove this, but with little success. Then, in 1966, and associated with his work on Goldbach’s conjecture, Chen Jingrun used sieve methods to prove that there are infinitely many prime numbers p for which is either a prime or an almost prime.
Among other more recent investigations were proofs that the twin prime conjecture is true if one is allowed to assume certain additional results. Then, suddenly and unexpectedly, a major breakthrough was made in June 2013 by the Chinese-born American mathematician Yitang Zhang. Using some of this earlier work, but without needing to assume any other results, he proved that infinitely many pairs of prime numbers differ by at most 70 million. This is a very far cry from the desired result, with 70 million instead of 2, but it was the first result of its kind and it created a whole cottage industry of mathematicians seeking to reduce the gap to something more manageable.
At first, the gap came down from 70 million to around 42 million, and then something rather remarkable happened. Mathematicians are used to writing up their research and then publishing it in polished form in journals, a process that can take a year or more: this means that much time may elapse before their results become widely known. But from around 2009 various mathematicians—notably, Tim Gowers from Cambridge and Terry Tao from Los Angeles—had proposed a more collaborative approach, known as the Polymath project, in which contributors from around the world could work on problems publicly by pooling their ideas, feeding in comments, and suggesting improvements. Advances could thereby be made and shared at a much faster rate, while individual contributors would still receive due credit for their ideas.
In June 2013 Tao initiated Polymath8, a project entitled ‘Bounded gaps between primes’, in which contributors were invited to improve Zhang’s result. Within a few weeks, with contributions from many people, the gap had decreased from 42 million to 387,620, and then to about 12,000, and after another month to 4680.
This was followed by a lull in activity, and new ideas were needed. By this time, James Maynard, who had gained his doctorate in Oxford and was then in Montreal, appeared on the scene with a different approach, also discovered independently by Tao. By the end of 2013 Maynard had reduced the gap to 600 and, as he wrote at the time:
I was surprised at how much time I ended up devoting to the Polymath project. This was partly because the nature of the project was so compelling—there were clear numerical metrics of ‘progress’ and always several possible ways of obtaining small improvement, which was continually encouraging. The general enthusiasm amongst the participants (and others outside of the project) also encouraged me to get more and more involved in the project.
At the time of writing, the current gap is 246. This is a massive improvement on the original 70 million—but there’s still quite a way to go before the twin prime conjecture is finally laid to rest.
The distribution of primes
How are the prime numbers distributed? Although they generally seem to ‘thin out’, the further along the list we proceed, they don’t seem to be distributed in a regular manner. For example, the hundred numbers just below 10 million include nine primes,
whereas the hundred numbers just above it include only two,
But although twin primes seem to crop up however far we go, we can also construct arbitrarily long strings of numbers that aren’t prime. To do so, we’ll use the factorial numbers n!, defined by
and in general,
Now, because the
