The Twin Prime Conjecture

         This weeks post will be covering a progression of the twin prime gap conjecture. The problem is faulty simple, and is only recently had a major breakthrough. Although there have been breakthroughs the problem remains unsolved. Firstly the problem itself will be covered. Secondly the order of progressions of the problem will be covered in this post.

         The twin prime conjecture is a fairly straightforward conjecture. It has been proven that there are an infinite number of prime numbers. The means that there is an infinite number of numbers that have no divisors other then one and itself. The twin prime conjecture focuses on the gap between prime numbers. There is one exception, but for the most part the smallest gap between two numbers is two. This is because every other number is even, so every other number is divisible by two. The only exception to that rule is, two, the only even prime number. If two prime numbers have a gap of two between them, then they are called twin primes. Primes with a gap of four are called cousin primes, and gaps of six are called sexy primes. The conjecture says that there is an infinite number of town primes. Meaning there is an infinite number of primes with only a gap of two between them. The origins of this problem are unknown. Some individuals say that the work comes from Euclid a couple thousand years ago. This could be very possible, but it was only formally written down a few hundred years ago, so it is at least that old. 

         There have been a number of breakthroughs in recent years towards this problem, but the first big breakthrough, that got the wheels turning happened within the last five years. From the University of New Hampshire a man by the name of Yitang Zhang released a paper putting a limit on the gaps. Before Yitang there was no approach to the problem, what he managed to do was say that there is an invite number of primes with gap N, N is between one and seventy million. To be clear he did not prove that the gap between primes are seventy million or less, but rather that there is an infinite number of primes with gap of seventy million. This breakthrough allowed for the first time a method of whittling down this seventy million to two. Yitang's result was the breakthrough, but it was not optimized. This means that though his result was seventy million, it was also understood that by tweaking the argument the bounds could shrink. This lead to a competition to see who could tweak the argument in such a way, to get the smallest bound. These developments lead to the gap being shrunk to 4,680, which is a big improvement, but not quite the end goal. Though the twin prime conjecture has not been proven, there have been more recent results that whittle down the gap number even more. Before the work of Yitang, there was an approach to solve the problem by mathematicians Goldston, Pintz, and Yildirim. Their work was important, but it required outside proofs, meaning that this works, but only if this other problem is solved. By sorting through these hiccups of the work of Goldstone, pints, and Yildirim, James Mayard, and Terrence Tao were able to whittle it down even further. Their method was completely different to Yitang, and both developed it simultaneously, and separately. Thus their work was named the Mayard-Tao method, and it brought the gap down to 256. 

         Even though this newly found method offers new insight into the problem, it has a flaw. The method used has a limit to it. If the method is completely optimized, it sill can not be used to prove the twin prime conjecture. Theoretically the Mayard-Tao method can only bring the gap down to there being an infinite number of primes with a gap of six. So even if the this method is completely optimized it is only a victory for the sexy primes. The win prime conjecture still needs a new approach, or new idea, to be solved.




Sources https://www.youtube.com/watch?v=vkMXdShDdtY
                  https://www.youtube.com/watch?v=QKHKD8bRAro
                  http://mathworld.wolfram.com/TwinPrimeConjecture.html