Terence Tao

Over time, it became more and more productive to think of the primes as if they were just generated by some god rolling dice all the time and just creating this random set.

And this allowed us to make all these other predictions.

So there's this still open conjecture in number theory called the twin prime conjecture, that there should be infinitely many pairs of primes that are twins, just as two apart, like 11 and 13.

We can't prove that.

And there are actually good reasons why we can't prove it.

But because of this statistical random model of the primes, we are absolutely convinced it's true.

We know that if the primes were sort of generated by flipping coins or something, that just by random charge, just like infinite monkeys at a typewriter, we would see twin primes appear over and over again.

And we have over time developed this very accurate conceptual model of what the primes should behave like based on statistics and probability.

But it's all mostly heuristic and non-rigorous, but extremely accurate.

So the few times when we actually can prove things about the primes, it has matched up with the predictions of what we call the random model of the primes.

So we have this conjectural concept framework for understanding the primes that everyone believes in.

And it's the same reason why we believe the Riemann hypothesis is true, why we believe that cryptography based on the primes is mathematically secure, things like that.

It's all part of this belief.

In fact, one reason why we care about the Riemann hypothesis is that if the Riemann hypothesis failed, we knew it was false, it means that it would

it would be a serious blow to this model.

It would mean there's a secret pattern to the primes that we were not aware of.

And I think we would very rapidly abandon any cryptography based on the primes, because if there was one pattern that we didn't know about, there's probably more.

And these patterns can lead to exploits in crypto.

And yeah, it's going to be a big, big shock.

So we really want to make sure that doesn't happen.