Twin primes are primes that occur is succession with one composite between them. For example, 3 and 5 are twin primes. They are primes in that they only have factors 1 and themselves, and then they are twins because they are separated only by one composite which is 4. The Twin Prime Conjecture claims that we can find this sort of pattern to infinity. No matter how far we go out on the number line, there will be another pair of twin primes to find. To view a really cheesy but fun video about this, check out the NOVA Science Now link.
Recently, the math world has exploded with discovery concerning this conjecture. Mathematicians had not been able to prove the Twin Prime Conjecture exactly as is, but they were able to prove that primes go on in a skippy pattern with more than one composite between them. Efforts are being made to narrow the gap, and hopefully eventually prove the gap of only one.
In May of 2013, it was revealed that Tom Zhang of the University of New Hampshire proved infinite pairs of primes that are 70 million apart. This seems like a big gap, but it was remarkable as it was the closest anyone had come. (It is definitely a smaller gap than the previous infinity.) To make it even more astounding, Professor Zhang was relatively unknown until this. He has been quietly working on his mathematics for years as he went about life and work in jobs such as sandwich maker for Subway.
The big reveal of Dr. Zhang’s discovery started a flurry of work. Dr. Terry Tao started a Polymath Project to enlist many volunteers to continue work on this project. The team narrowed down the gap from 70 million to 4,680 by July.
As of September, James Maynard had reduced the gap to 600.
In November, Thomas Engelsma says he reduced the gap to 576.
And the work goes on!