Terence Tao

But you could easily, if you wanted to, redact the primes to get rid of these twins.

The twins, they show up, and there are infinitely many of them, but they're actually reasonably sparse.

Initially, there's quite a few, but once you go to the millions, trillions, they become rarer and rarer.

And you could actually just... If someone was given access to the database of primes, you just edit out a few primes here and there, they could make the twin-prime conjecture false by just removing 0.01% of the primes or something.

Just well-chosen to do this.

And so you could present a...

censored database of the primes, which passes all of the statistical tests of the primes.

It obeys things like the prime number theorem and other things about the primes, but doesn't contain any twin primes anymore.

And this is a real obstacle to the twin prime conjecture.

It means that any

proof strategy to actually find trend primes in the actual primes must fail when applied to these slightly edited primes.

And so it must be some very subtle, delicate feature of the primes that you can't just get from like aggregate statistical analysis.

Okay, so that's out.

Yeah.

On the other hand, arithmetic progressions has turned out to be much more robust.

You can take the primes, and you can eliminate 99% of the primes, actually.

And you can take any 99% you want.

And it turns out, and another thing we proved, is that you still get arithmetic progressions.

Arithmetic progressions are much, you know, they're like cockroaches.

Of arbitrary length.