Terence Tao

So you can make a conspiracy rule out one of these, but once you have like 50 of them, it turns out that you can't rule out all of them at once.

It just requires too much

energy somehow in this conspiracy space.

So it's ultimately based on what's called the pigeonhole principle.

So the pigeonhole principle is a statement that if you have a number of pigeons, and they all have to go into pigeonholes, and you have more pigeons than pigeonholes.

then one of the pigeon holes has to have at least two pigeons in it.

So there has to be two pigeons that are close together.

So, for instance, if you have 100 numbers and they all range from 1 to 1,000, two of them have to be at most 10 apart.

Because you can divide up the numbers from 1 to 100 into 100 pigeon holes.

Let's say we have 101 numbers.

If we have 101 numbers, then two of them have to be a distance less than 10 apart because two of them have to belong to the same pigeon hole.

So it's a basic feature of a basic principle in mathematics.

So it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out, that fewer and fewer numbers are prime.

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.