## Oct 19 Twin Primes

First off, let’s just get out of the way: there are an infinite number of primes. Skip ahead a paragraph if this is old news to you. We have known this since Euclid ~300 BCE, which is considerably before we knew how to find primes; which is so very typical of the Greek mathematicians. The proof goes like this: assume there are a finite number primes. We call name them, p(1), p(2) up to the biggest one p(n). But if we multiply them all together and add 1, then we get a number which is 1 more than a multiple of two and one more than a multiple of 3 and … and 1 more than a multiple of p(n). So it must be prime itself, so our original assumption was wrong, so there are an infinite number of primes.

A twin prime is one of a pair of primes that lie only two apart from each other. Notice that because almost all primes are odd then this is as close as they can be apart from right at the start. So 3 and 5 are twin prime, as are 11 and 13, but 23 is not a twin prime because neither 21 or 25 are prime. Since primes get rarer as we go along it means that twin primes become very rare. We don't know whether there are an infinite number of them and it is a famous unsolved problem in mathematics. I'm going to spend the rest of the article trying to explain some of the work we have done towards this goal.

However we do know how many sets of prime triplets (p, p+2 and p+4 all prime) there are. Have a go at working it out yourself. Answer at the bottom of the article.

Many of the mathematical greats have attempted the twin prime problem over time, but on the 14th May 2013 (about a month before I qualified to be a teacher) we made our first progress ever. A mathematician named Zhang proved that as you go along the number line there will always be another pair of primes which are at most 70,000,000 apart. Notice that that isn't the same as saying that the next prime number is at most 70 million away from the last, the gaps can get much bigger than that. It was the first time we had put any bound on these gaps at all.

This number 70,000,000 wasn't optimised at all, it took a whole lot of upper bounds without seeking to minimise this number. Looking at Zhang’s paper another mathematician named Lewko managed to reduce the bound to 63,374,611 a week later on 21st May. However once the idea of lowering the bound was established the method was published in an open source way so that mathematicians from around the world could help bring the bound down. If the bound could be reduced to 2 then we would have solved the Twin Prime Conjecture.

As more people in the mathematical community caught on to the idea the rate of improvement increased. It was actually really exciting as people made lots of wikis on the fly with the different techniques that mathematicians were using. At the peek there was a new lower bound every half an hour and I was following it all through r/math on reddit. Here’s a link to a timeline of improvements over time (it is a bit complicated, but you are looking at the third column for entries without m=number or [EH]). By 14th April 2014 we had plateaued at 246 (6 if we assume another theorem that we think is true and we think we can probably prove easier).

However for now we think that we need another completely new idea to bring the limit all of the way down to two. I, for one, enjoy that work on a famous unsolved problem from centuries ago is still moving forwards and this switch to open source seems to be the natural extension to having huge groups of mathematicians already working on problems. Gone are the old days of having a lone mathematician go into their study and solve something by themselves; in short, all the low hanging fruit has been picked. Maths has gone collaborative.

Also, on the topic of primes, behold this beauty.

Solution

For the triplet primes there is only one set, 3, 5 and 7. It is important to realise that in any set of three numbers: n, n+2 and n+4 there is exactly one multiple of 3. There is only one number which is both a multiple of 3 and a prime, namely 3 itself so the set above is the only one. I quite like that the jump from the twin prime problem (so far unsolved by all humans ever) and the triplet prime problem (relatively easy) is so great despite almost identical set ups.