Terence Tao

Yes.

Yes.

That's crazy.

Yeah.

Yeah.

But it's again like, it's an infinite monkey type phenomenon.

For any fixed length of your set, you don't get arbitrary length progressions.

You only get quite short progressions.

But you're saying twin prime is not an infinite monkey.

I mean, it's a very subtle one.

It's still an infinite monkey phenomenon.

If the primes were really genuinely random, if the primes were generated by monkeys, then yes, in fact, the infinite monkey theorem would... Oh, but you're saying that twin prime, you can't use the same tools.

Well, we don't know.

We believe the primes behave like a random set.

And so the reason why we care about the Trim-Half Conjecture is it's a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set.

Random versions of the primes we know contain twins, at least with 100% probability, or probably tending to 100% as you go out further and further.

Yeah, so the primers, we believe that they're random.

The reason why athmic progressions are indestructible is that regardless of whether your set looks random or looks structured, like periodic, in both cases, athmic progressions appear, but for different reasons.

And this is basically all the ways in which there are many proofs of these sort of athmic progression epitheliums, and they're all proven by some sort of dichotomy where your set is either structured or random, and in both cases, you can say something, and then you put the two together.

But in twin primes, if the primes are random, then you're happy, you win.