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:
- Twin prime, Wikipedia, the free encyclopedia
- Prime number theorem, Wikipedia, the free encyclopedia
- Several Proofs of the Twin Primes Conjecture
- Bounded gaps between primes, Polymath1Wiki
- The Work of Pierre Deligne, N.M. Katz pdf
- Progress on the Twin Primes Conjecture, EvolutionBlog
- Further Progress on the Twin Primes Conjecture, EvolutionBlog
- Twin Prime Conjecture, from Wolfram MathWorld
- Twin prime conjecture
- Yitang Zhang, twin primes conjecture: A huge discovery about prime numbers, and what it means for the future of math.
- Is math a young man’s game?
- twin prime conjecture, number theory, Encyclopedia Britannica
- The Top Twenty: Twin Primes
- The Twin Prime Hero, Issue 5: Fame, Nautilus
- Mathematicians Team Up on Twin Primes Conjecture, Simons Foundation
- Yitang Zhang Proves ‘Landmark’ Theorem in Distribution of Prime Numbers, Simons Foundation
- What is the smallest prime? arxiv 1209.2007
- MAJOR ARCS FOR GOLDBACH’S PROBLEM, H. A. HELFGOTT, pdf
- Small gaps between primes, arxiv 1311.4600
- Primes in Tuples I, arxiv math/0508185
- Bounded gaps between primes! at E. Kowalski’s blog
- Chance versus Randomness, Stanford Encyclopedia of Philosophy
- I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Secret Blogging Seminar
- Introduction to twin primes and Brun’s constant computation
- Mathematician proves there are infinitely many pairs of prime numbers less than 70 million units apart
- An old mathematical puzzle soon to be unraveled?
- Polymath8b: Bounded intervals with many primes, after Maynard, What’s new
- Polymath8: Writing the paper, What’s new
- Marker lecture III: “Small gaps between primes”, What’s new
- Don’t prematurely obsess on a single “big problem” or “big theory”, What’s new
- Simons Lecture I: Structure and randomness in Fourier analysis and number theory, What’s new
- The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, What’s new
- Polymath proposal: bounded gaps between primes, The polymath blog
- sieving admissible tuples
- The Paradox of the Proof, Project Wordsworth
Must get to understanding Yitang Zhang solution…