Closing the Gap between Prime Numbers

Occasionally, a major mathematical discovery comes from an individual working in isolation, and this gives rise to great surprise. Such an advance was announced by Yitang Zhang six years ago. [TM161 or search for “thatsmaths” at irishtimes.com].

Yitang-Zhang-Colour

Yitang Zhang

After completing his doctorate at Purdue in 1991, Zhang had great difficulty finding an academic position and worked at various humdrum jobs. Eventually, he managed to obtain a lectureship at the University of New Hampshire.

At a seminar in Harvard in April 2013, Zhang proved that there are infinitely many pairs of prime numbers differing by at most 70,000,000. That is, there are unlimited pairs of primes where the gap between them is less than seventy million.

Zhang’s result was ground-breaking. He was in his late 50s, smashing the stereotype that brilliant new advances are made only by young mathematicians. This was the first time that anyone had managed to prove any result along these lines.

The Atoms of the Number System

Prime numbers are indivisible; they are the atoms of the number system, and cannot be split into products of smaller numbers. Thus, 11 is a prime while 12, being 3 times 4, is not. The first few primes are 2, 3, 5, 7 and 11. Euclid, famed for The Elements, his book on geometry, also wrote on number theory and showed, by a beautiful argument, that the set of prime numbers is infinite.

As numbers get larger, there are more possible divisors, so the chance of a randomly chosen number being prime decreases. Thus, the average difference, or gap, between successive primes grows. However, their distribution is far from regular, and they tend to cluster in unpredictable ways. Pairs of primes differing by 2 can easily be found: (5, 7), (17, 19), (59, 61) and (101, 103) are examples. It is not too difficult to show that, excepting (3, 5), the average of two twin primes is a multiple of 6. Twin primes become increasingly rare as we examine larger ranges. However, it seems that there is an unlimited number of prime pairs. This Twin Prime Conjecture has never been proved.

The Twin Prime Conjecture

Zhang’s 70,000,000 was far from the gap of 2 required to prove the Twin Prime Conjecture. The question arose: can we close the gap? Within a month, it had been reduced to 42 million. By mid-June 2013 it was 400,000 and by the end of June just 12,000. Six months after Zhang’s result appeared, James Maynard, who had recently completed a PhD at Oxford, reduced the gap to 600. That is, he proved that there exists an infinite set of pairs of prime numbers that differ by at most 600. The bound has since been reduced to 246.

Many prime number pairs have been found using computers, but this approach will never prove that there are infinitely many pairs. We have made substantial progress, much of it in the Polymath-4 Project. But, for now, the Twin Prime Conjecture remains unproven. The quest to close the gap is ongoing.

There are many other questions about prime numbers that are easy to state but so-far impossible to prove. The set of pairs of primes of the form (p, 2p+1) are known as Germain primes, after the mathematician Sophie Germain. The pair (11, 23) is an example. It is not known whether this set is infinite or not. Neither is it known whether the set of primes that exceed a perfect square by one (p = n2+1) is finite of infinite. But the Twin Prime Conjecture is the most famous unsolved problem in this field.

Sources

Vicky Neale, 2017: Closing the Gap. Oxford Univ. Press. 164pp. ISBN: 978-0-198-78828-7.


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