We're 99.9% sure this pattern is true, but no one can prove it

- [Derek] On the morning of the 17th of April, 2013, the journal Annals of Mathematics received a curious email.

It claimed to contain a 50 page proof relating to one of the oldest unsolved problems in mathematics.

A problem that great mathematician Edmund Landau called unattackable. But this proof didn't come from a famous professor, it came from an unknown.

(bright music) Someone who had once spent years working at a subway restaurant. - So, they're like, okay, surely this isn't gonna work out, but whatever.

You know, we'll send it to a referee. - [Derek] They expected to find a mistake in an afternoon, but they didn't.

So, they went through it again, closely studying the fragile parts where a proof like this normally falls apart, but still nothing.

Soon they realized they were witnessing a breakthrough. - Oh, damn, he did it. - So, how did he do It?

What did the experts miss and what was the problem he was working on? He was working on a new way to attack one of the hardest unsolved problems in number theory, the twin prime conjecture.

The twin primes are prime numbers separated by just one number, like 11 and 13 or 17 and 19. As you go up, the number line, primes become rarer and twin primes become rarer still.

But the twin prime conjecture claims that there are infinitely many of them you never run out. But is it true?

Well, one way to approach this problem is to look at the gaps between consecutive primes as you go up the number line.

(bright music) Now, at first they seem chaotic, but if you average them out, a clear trend emerges.