This week, That’s Maths in the Irish Times ( TM022 ) reports on two exciting recent breakthroughs in prime number theory.
The mathematics we study at school gives the impression that all the big questions have been answered: most of what we learn has been known for centuries, and new developments are nowhere in evidence. In fact, research in maths has never been more intensive and advances are made on a regular basis.
Two Problems solved in May 2013
During the past month, two major results in prime number theory have been announced. Euclid showed, some 23 centuries ago, that there is an infinitude of primes, but many fundamental questions about their properties remain unanswered. For example, prime pairs like 17 and 19, differing by 2, may be finite or unlimited in number. No-one has the answer to this ‘Twin Prime’ problem.
Another puzzle surrounds the splitting of even numbers into sums of primes, like 10 = 3 + 7. Is this possible for every even number? Christian Goldbach thought so and said as much in 1742 in a letter to his friend Leonhard Euler, the great Swiss mathematician. Euler responded that he regarded Goldbach’s conjecture as “virtually certain, though I cannot prove it”. And no-one has proved it since.
Why should we worry? Number theory is the purest of pure mathematics, remote from our daily cares. How could it have any relevance to practical life? In fact, the properties of prime numbers underlie all modern cryptography, which is vital for the integrity of online communications and the security of internet financial transactions.
Yitang Zhang’s Result
Last month, Yitang Zhang of the University of New Hampshire sent a paper to the pre-eminent journal Annals of Mathematics, claiming that there are an infinite number of prime pairs whose separation is less than a fixed constant. The constant is huge, about 70 million, a long way from 2, the value needed to prove the twin prime problem. Still, it is a dramatic breakthrough, and ways will soon be found to reduce the separation constant. Zhang’s paper was fast-tracked for review and within three weeks one referee had described it as “first-rate”. Zhang presented his results to a capacity audience in Harvard on 13 May.
Harald Helfgott’s Result
On that very same day, Harald Helfgott of Ecole Normale Supérieure in Paris posted a 133 page preprint proving a weak form of Goldbach’s conjecture: every odd number from 7 upwards is the sum of three primes. This would follow from Goldbach’s statement about even numbers, but the argument does not work the other way round, so the original conjecture remains open. Here is Helfgott’s paper on arXiv.
There is cast-iron evidence that Goldbach was right: all even numbers up to 4×1018 have been shown to be sums of two primes, but the greatest mathematicians have been unable to prove that the assertion is true in all cases. The 270 year old conjecture remains open.