Terence Tao

One, two, three, four, five.

Or you can take the prime number, if you want to generate multiplicatively, you can take all the prime numbers, two, three, five, seven, and multiply them all together.

And together, that gives you all the natural numbers, except maybe for one.

So there are these two separate ways of thinking about the natural numbers from an additive point of view and a multiplicative point of view.

And separately, they're not so bad.

So any question about the natural numbers that only involves addition is relatively easy to solve.

And any question that only involves multiplication is relatively easy to solve.

But what has been frustrating is that you combine the two together.

And suddenly you get an extremely rich, I mean, we know that there are statements in number theory that are actually as undecidable.

There are certain polynomials in some number of variables.

Is there a solution in the natural numbers?

And the answer depends on an undecidable statement, like whether the axioms of mathematics are consistent or not.

But even the simplest problems that combine something multiplicative, such as the primes, with something additive, such as shifting by two.

Separately, we understand both of them well, but if you ask, when you shift the prime by two, how often can you get another prime?

It's been amazingly hard to relate the two.

Right.

Yeah, so what we've realized because of this type of research is that different patterns have different

levels of indestructibility.

So what makes the twin prime problem hard is that if you take all the primes in the world, you know, 3, 5, 7, 11, so forth, there are some twins in there.

11 and 13 is a pair of twin primes, so forth.