Terence Tao

So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, this process that goes up if you're odd and down if you're even, it eventually goes down to earth all the time.

No matter where you start with this very simple algorithm, you end up at one.

And you might climb for a while, come down.

Yeah, if you plot it, these sequences, they look like Brownian motion.

They look like the stock market.

They just go up and down in a seemingly random pattern.

And in fact, usually that's what happens, that if you plug in a random number, you can actually prove, at least initially, that it would look like a random walk.

And that's actually a random walk with a downward drift.

It's like if you're always gambling on a roulette at the casino with odds slightly weighted against you.

So sometimes you win, sometimes you lose.

But over, in the long run, you lose a bit more than you win.

And so normally your wallet will go to zero if you just keep playing over and over again.

Yes.

So the result that I proved, roughly speaking, is that statistically, like 90% of all inputs would drift down to, maybe not all the way to one, but to be much, much smaller than what you started.

So it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started.

That's kind of like the result that I proved.

Well, the problem is that I used arguments from probability theory.

And there's always this exceptional event.

So in probability, we have these low, large numbers, which tells you things like if you play a casino with a game at a casino with a losing expectation, over time, you are guaranteed, almost surely, with probability,

probably probably as close to 100 as you wish you're guaranteed to lose money but there's always this exceptional outlier like it is mathematically possible that even in the game is the odds are not in your favor you could just keep winning slightly more often than you lose very much like how in navier stokes it could be you know most of the time your waves can disperse there could be just one outlier choice of initial conditions that would lead you to blow up