Back in April, Zhang Yitang came up with a result that is a major step toward proving the twin prime conjecture that there are infinitely many primes p for which p + 2 is also prime.
ELI5 attempt at the twin prime conjecture
Think of cookie parties.
If you have 100 cookies, you could have a cookie party:
- by yourself (you get all the cookies)
- with two people (each person gets 50 cookies)
- with four people (each person gets 25 cookies)
- with five people (each person gets 20 cookies)
- with ten people (each person gets ten cookies)
- with 20 people (each person gets five cookies)
- with 25 people (each person gets four cookies)
- with 50 people (each person gets two cookies)
- with 100 people (each person gets one cookie)
If you're the only person at your party, it's a sad party.
If everyone at the party gets only one cookie, it's a sad party.
If someone gets more than someone else, it's a sad party.
You don't want your party to be sad, so you have to be careful to have the right number of people to share your cookies.
If you have two cookies, or three, or five, or seven, or eleven, then it's not possible to have a happy party. There's no "right number of people."
People used to wonder whether you could be sure to have a happy party if you just had enough cookies. A famous person named Euclid figured out that, no matter how many cookies you had, even if it was, like, more than a million, you might be unlucky and have a sad number of cookies.
If it's a birthday party, the birthday kid's mom might give the birthday kid an extra cookie. (Or they might get something else instead.) That would be OK.
If it's a birthday party, then, yes, you can be sure to have a happy party if you just had enough cookies. In fact, even three cookies would be enough; you could have the birthday kid, and one friend; they would each have one cookie, and the birthday kid would get the extra one.
But Sam and Jane have a problem. They're twins, and they always have the same birthday. One year they had 13 cookies, and it was a big problem. 13 is a sad number. Even if they both had an extra cookie, that would leave 11, and that is still a sad number.
(If you allow the birthday kid to have two extra cookies, that would leave nine; they could invite one more person, give everyone three cookies, and then Sam and Jane could each have two extras. But this is not a happy party because the guests will get upset that the birthday kids got two extra cookies. I mean, come on!)
Sam and Jane wondered whether they could be sure to have a happy party if they just had enough cookies.
So they asked their mom, who is, like, super smart.
But even she didn't know.
In fact, no-one knows. They don't think so. But they're not, like, super-sure.