Twin Primes and the Ternary Goldbach Conjecture

English: The Sieve of Eratosphenes finds prime...


Just like buses… exactly just like buses… you wait for one and two come all at once. That is exactly what I thought when I heard about the two results in number theory that came up almost at the same time. One by Yitang Zhang, of the University of New Hampshire in relationship to twin primes, and the other one by Harald Andrés Helfgott of École Normale Supérieure – Paris about the ternary Goldbach conjecture. So what are these things?




Twin Primes




Well Yitang Zhang’s paper in Annals of Mathematics, entitled “Bounded gaps between primes” deals the question as to how close  two prime numbers are. The Twin Prime Conjecture states that there are an infinite number of primes p and q that are as close as possible: p-q=2. (ahem… I know, some of you are already raising your hands saying that 2 and 3 are closer, but that can only happen once…). Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. And Zhang has tackled the problem head on by showing that there are an infinite number of primes p and q such that p-q=2N, where N is a bounded constant. The new result therefore shows that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. It may seem that 70 million is way to large a number, not compared to infinity! This means that the gaps between consecutive numbers don’t keep growing forever.




The Ternary Goldbach Conjecture




Keeping with prime numbers, they are indeed of much interest as they can be seen as the “atoms of arithmetic” as other number can be expressed as factorisation of prime numbers. In that sense, prime numbers are intimately related to multiplication, but there are additive properties that they do have. The Goldbach conjecture proposes that every even number is the sum of two primes and Harald Andrés Helfgott settled the matter of a weaker version of the conjecture. You can have a look at his paper entitled “Major arcs for Goldbach’s problem” here. The statement that Helfgott  provided a proof for us that every odd integer n>5 is the sum of three primes. Although Helfgott’s paper has not yet been formally published or peer-reviewed, it has been endorsed by Terrence Tao, who was close to resolving the problem last year.