Ciro Santilli @cirosantilli 35

4 squares are sufficient by Lagrange's four-square theorem.

3 is not enough by Legendre's three-square theorem.

The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem and

The elliptic curve group of all elliptic curve over the rational numbers is always a finitely generated group.

The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.

For this reason, the rank of an elliptic curve over the rational numbers is always defined.

TODO example.

Mordell's theorem guarantees that the rank (number of elements in the generating set of the group) is always well defined for an elliptic curve over the rational numbers. But as of 2023 there is no known algorithm which calculates the rank of any curve!

It is not even known if there are elliptic curves of every rank or not: Largest known ranks of an elliptic curve over the rational numbers, and it has proven extremely hard to find new ones over time.

TODO list of known values and algorithms? The Birch and Swinnerton-Dyer conjecture would immediately provide a stupid algorithm for it.

math.mit.edu/classes/18.783, Wow, good slides! Well organized site! This is a good professor! And brutal course. 25 lectures, and lecture one ends in BSD conjecture!

Some points from math.mit.edu/classes/18.783/2022/LectureSlides1.pdf:

Their status is a mess as of 2020s, with several systems ongoing. Long live the "original" collegiate university!

web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:TODO why this non standard formulation?

Moodle instance of the Mathematical Institute of the University of Oxford.

Has a mixture of open access and closed access. But at least it can have open access unlike the in-house systems such as Canvas where everything is necessarily paywalled!

Sometimes things appear open but don't show any meaningful content if you are not logged in, which is annoying.

But at least it gives a clear public course list, thing that certain departments (cough Department of Physics of the University of Oxford cough).

The organization is a bit crap, when you expand e.g. C Michaelmas term it shows nothing, just a search.

The way to go is via the year year categories e.g. "Year 2022-23": courses.maths.ox.ac.uk/course/index.php?categoryid=734. Term splitting is annoying, but one can stand it.

There seems to be no way to list all versions of a single course across multiple years besides just doing a search e.g.