The twin primes conjecture asserts that there are infinitely many pairs of twin primes — prime numbers that have only one number between them, like 11 and 13, or 17 and 19.
While the twin primes conjecture has not been proven, mathematicians have, over the last several months, made significant progress in showing that there are infinitely many pairs of primes that are, at most, some fixed distance from each other.
Erica Klarreich at Quanta magazine has a wonderful write up of the recent research.
The prime numbers — numbers that are divisible only by themselves and one — have always been a major subject of study in mathematics.
One particular aspect of the primes that has fascinated mathematicians throughout the centuries is their distribution — where primes fall on the number line. We know that the prime numbers become over all rarer as numbers get larger, but do primes sometimes cluster together, or do the gaps between consecutive prime numbers get larger and larger?
The extreme case of primes clustering together is exemplified in the twin primes conjecture.
Most mathematicians have a sense that the twin primes conjecture should be true — the positioning of the prime numbers appear to be more or less random, even though on average the gaps between primes get larger, and if one has an infinitely long list of random odd numbers, we should have an infinite collection of pairs in our list. If at some point, prime numbers are always more than two numbers away from each other, we have a non-random aspect to their distribution that goes against this intuition.
2013 has been an exciting year for this line of research. In May, University of New Hampshire mathematician Yitang Zhang showed that there are infinitely many pairs of primes that are no more than 70,000,000 numbers away from each other. This was the first time such an upper bound was established.
Over the summer, a number of mathematicians collaborated to refine Zhang's work and reduced that bound to around 4400.
Now, James Maynard, a researcher at the University of Montreal, has submitted a paper with a new approach. He has shown that there are infinitely many pairs of primes no further than 600 numbers away from each other.
The speed with which this type of research has developed this year is extremely impressive, and we are getting closer to a better understanding of the building blocks of number theory.