Wednesday, 22 May 2013

A surprise in the world of prime numbers

The world of mathematics is buzzing with the news of a breakthrough in the theory of prime numbers by a virtually unknown mathematician named Yitang Zhang.  Prime numbers are numbers like 3, 5, 23, 97 which cannot be expressed as a product of two smaller numbers.  Prime numbers have been studied since the time of the Ancient Greeks and about 23 centuries ago the famous geometer Euclid discovered that there are an infinite number of them.  Since that time the study of prime numbers has produced some of the deepest results in the whole of mathematics yet these results are often very easy to understand (but the reasons the results are true are another matter).  

It is easy to see that 2 and 3 are the only consecutive prime numbers (because if you have two consecutive numbers one of them must be even but 2 is the only even prime number).  What about prime numbers differing by 2?  Here there are more: (3, 5), (5, 7), (11, 13) for example and there are many other so-called twin primes.  But are there are an infinite supply of twin primes?  Despite hundreds of years of research we don't know the answer to this innocuous question.

What about pairs of prime numbers that differ by 4 (like (3,7) or (19, 23))?  We don't know if there are infinitely many such pairs either.  And if you replace 4 by any other even number at all we still don't know.  For example, we don't know whether there are infinitely many prime pairs that differ by 30 say (like 31 and 61).

Enter Yitang Zhang.  In his 50's he is definitely beyond the age where most mathematicians make their mark and until now he has been practically unknown.  After he got his PhD in 1992 he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.  But he never gave up doing mathematics and eventually was appointed at the University of New Hampshire.  There he pursued an unremarkable research career with no publication since 2001 but he was loved by his students apparently because he set easy exams.  Now he has burst onto the world's mathematical stage with a result that has surprised all the experts.  He has proved that there is some number k that "works" for prime pairs.  We don't know that k=2 or k=4; all we know is that k is less than 70 million.  And for this k, whatever it is, Zhang has proved that there are infinitely many pairs of prime numbers that differ by k.

The reason that this has excited mathematicians is that no result like this has ever been proved before and it gives some hope that the original prime twin problem might eventually be solved.  It is also a surprise result in that it hardly ever happens that a giant step like this is taken by such an obscure mathematician.  When it has happened before (such as for the Indian genius Ramanajuan) the newcomer is usually much younger since mathematics at its most creative is usually a young person's metier.

Zhang's theorem probably won't have much practical use.  If you like problems about ages and birthdays here's a consequence that might be appealing.  Somewhere, sometime, there were two mammals of different ages and there will be an infinite number of years when both their ages are prime numbers.  Might there be two human beings with this property?  We don't know because humans haven't been around for 70 million years whereas mammals have.

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