We all know what prime numbers are – they are numbers that are divisible only by themselves and by the number one. Here are a few primes: 1, 2, 3, 5, 7, 11, 13, 17, 19 and so on. For small numbers, it is easy to check whether they are prime numbers and to get a handle on their properties. But it is not so easy when they are really large. From experience, people develop beliefs or “conjectures” about the properties of large primes.
One thing that people observe is that while prime numbers occur frequently in a series when they have small values, they appear to get spaced out further and further as we go to very large numbers. So do they ultimately disappear at some point?
This is where a famous conjecture called the twin prime conjecture comes in.
Twin primes
This conjecture was first documented by Alphonse de Polignac in 1849 – that’s 165 years ago. The belief is that there are pairs of primes separated only by 2 units (i.e. n, n+2: for example 3 and 5 or 17 and 19) and that there are infinitely many of them. Now, this is a conjecture – which means it is a statement which has not been strictly proved using existing mathematical theorems. People trying to solve it on and off throughout history, but it has eluded people for years
Then, in April 2013, came along a Chinese mathematician Yitang Zhang, who proved a result which is very close to the twin prime conjecture. He proved that there are infinitely many such pairs of primes and the separation between them is a number that is not more than 70 million. That is, that there are infinitely many pairs of the form (n, n+N), and he had found an upper bound on their separation N. According to him, N cannot be more than 70 million.
Shrinking gap
Do you think 70 million is too huge a number as compared to 2? But the important thing here was that Yitang Zhang proved that there are infinite pairs of prime numbers that differ only by a finite number.
Since his discovery, there has been rapid progress on Prof. Zhang’s result.
Independently, James Maynard, a post-doctoral fellow and Terence Tao, a Fields medallist, have come up with proofs that the bound is not as high as 70 million, but only 600.
Tao also put his work on a blog – The Polymath Problem 8 – and crowdsourced for a solution. Many mathematicians got into it and due to their joint effort they brought down the gap to a mere 252.
They believe, if you assume a further conjecture (which hasn’t been proved yet) you can bring down the bound to 6 or so…
What about the twin prime conjecture?
But to bring down the gap to two and prove the twin prime conjecture — is that feasible? Mathematicians believe that however will require one more piece of the puzzle to solved — one more discovery or breakthrough is needed to make that final leap.
Still these are exciting times for mathematicians in Number Theory, for what has only been a belief for centuries is now emerging as a proven fact!
