Infinitely many twin primes: a reality

Take a closer look at the following numbers:

  • 3, 5
  • 5, 7
  • 11, 13
  • 17, 19
  • 29, 31
  • 41, 43
  • 59, 61
  • 71, 73
  • 101, 103
  • 107, 109
  • 137, 139

What can you say about them? First of all, all of them are prime numbers.

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Second of all, there is a gap of 2 between each pair of two prime numbers. It turns out that 2 prime numers that are just 2 apart are called twin primes.

A twin prime is a prime number that has a prime gap of two, in other word, differs from another prime number by two, for example the twin prime pair (41, 43).

If you keep writing twin prime numbers out, you will notice the general tendency of gaps between adjacent primes to become larger. A very natural question arises: do twin prime numbers continue for ever - i.e. is there an infinite number of twin primes?

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime.

Yitang Zhang, a lecturer in mathematics at the University of New Hampshire, came up with an answer to the twin prime conjecture. He showed that there exists a number N, that is at most 70 million, such that there are infinitely many prime pairs that differ by N. Covered in the news already, other mathematicians have stepped in. They were able to lower N down to 4680.

Although as of yet there is no paper from Zhang himself, James Maynard has submitted a description (“Small gaps between primes”, arXiv:1311.4600 [math.NT]) of the method that was used to get down to 4680.

Other interesting resources:

Must get to understanding Yitang Zhang solution…