Terence Tao

But it turns out that there's a way to assign weights to numbers.

So there are numbers that are kind of almost prime, but they don't have no factors at all other than themselves and one, but they have very few factors.

And it turns out that we understand almost primes a lot better than we understand primes.

And so, for example, it was known for a long time that there were twin almost primes.

This has been worked out.

So almost primes are something we can't understand.

So you can actually restrict the attention to a suitable set of almost primes.

And whereas the primes are very sparse

overall, relative to the almost primes, they actually are much less sparse.

You can set up a set of almost primes where the primes have density like, say, 1%.

And that gives you a shot at proving, by applying some sort of original principle, that there's pairs of primes that are just only 100 apart.

But in order to prove the twin-prime conjecture, you need to get the density of primes inside the almost primes up to a threshold of 50%.

Once you get up to 50%, you will get twin primes.

But unfortunately, there are barriers.

we know that no matter what kind of good set of almost primes we pick, the density of primes can never get above 50%.

It's called the parity barrier.

And I would love to find, yeah, so one of my long-term dreams is to find a way to breach that barrier.

Because it would open up not only the twin-prime conjecture, the go-back conjecture, and many other problems in number theory are currently blocked because our current techniques would require going beyond this theoretical parity barrier.

It's like going past the speed of light.

Oh, yeah.