Leonhard Euler

And so this was exactly the bootstrap idea.

And this very simple formula, which is Euler's beta function, which is some ratio of gamma functions, that gave an explicit realization of the bootstrap idea.

And it was a simple, well-known mathematical function as a good first approximation.

This would be sort of the leading term in some perturbative description.

It would take longer than the age of the universe to factor a large RSA key by brute force, trying every possible factorization, one after another, with even the most powerful computers available today.

Through additional mathematical operations involving something called Euler's totient function and modular arithmetic, you generate additional values that become part of the public and private keys.

The public key consists of n and another value called e, the encryption exponent, typically a small number like 65537, chosen for efficiency reasons.

The private key consists of N and another value called D, the decryption exponent.

The new y coordinate is then y sub 0 plus h times f of 0 y sub 0, and we can use this as our new starting point and repeat the process, drawing a new tangent line, extrapolating the change in y, and adding to our current approximation.

In general, we can write down what is known as Euler's method, x sub n plus 1 equals x sub n plus our step size h, and y sub n plus 1 equals y sub n plus h times f of x sub n y sub n.

This allows us to compute a whole sequence of y values starting from our initial condition, and progressively approximating further and further into our solution, as long as our step size H is small enough to not stray too far from whatever the true solution may be.

What if we weren't given an initial condition? In that case, we wouldn't have a fixed starting point, but we can still make some progress.

For example, notice that along the line x equals 2, all of the tangent lines have the same gradient.

We can therefore see that all of the solution curves will simply be vertical translations of each other which correspond to choosing different initial conditions.

One issue with Euler's method is that you need an extremely tiny step size h in order to get accurate results, which makes computations slow and susceptible to rounding errors.

This motivated the development of better numerical methods, so let's take a look at some of those now.

I'm looking at the tool right now.

It's finite time blow up for the Euler equation and finite time blow up for Navier-Stokes.

And the Navier-Stokes one is 157 pages and the Euler one is 56 pages of relatively dense argumentation like it's not horribly written but it's still something that i think like people will need a while to really settle and also like the the lean proof is not written in a form that would go into into math flip which is like this highly curated base of um existing proofs in a way that can be reused basically

right and

so and also what um what's important and we were doing like a preparatory discussion for for this before um i was trying to explain that when you do an actual proof like when you do a mathematical analysis what it means is that you're wrapping your head around something that is actually too big to fit into your head.

You know, you run out of letters pretty quickly.

I like to do it in lectures because actually Euler did that in his papers.

He would always, you know, instead of using indices one, two, three, he would use A for the first coefficient and B for the second and so on.

And he got a lot of mileage out of that.