290-Theorem Information
Description and Preprint:
The proof of the 290-Theorem was announced in Fall 2008, and is a joint project with Manjul Bhargava at Princeton University. The proof involves some fairly elaborate computer computations for which I have written specialized software in C++, Magma, and Python/Sage. Some of these computations (involving modular symbols) were originally performed by William Stein using floating point numerical linear algebra libraries, and these have been rechecked using exact rational arithmetic to ensure their numerical accuracy. The datafiles for the exact rational linear algebra computations are available here, and the paper should appear in the near future. For now, the original preprint can be found here. Also here is a talk I gave at Rutgers in Fall 2013 discussing the 290-Theorem and the techniques and ideas used in its proof.
Related Work and Publicity:
- “All Square” 2006 Science News Article
- “Quadratic Forms over $\mathbb{Z}$, from Diophantus to the 290-Theorem” by Alexander J. Hahn
- The 2008 Stanford Thesis of Yong Suk Moon, “Universal quadratic forms and the 15-theorem and 290-theorem“
- “Quadratic Forms Representing all Odd Integers” paper by Jeremy Rouse
- UCLA Course 290B on Quadratic Forms and the 290-Theorem
- Bhargava’s 2014 Fields Medal Citation, and the related
- Conway’s Princeton Colloquium [Comments on 290-Theorem, Abstract]
- “2014 Fields Medals” in Oct. 2014 Notices of the AMS
- “Why is 290 so special?” Math Blog
- “Manjul Bhargava: Revealing Numbers” Plus Magazine
Technical Information and Data:
The escalator data used in the proof of the 290-Theorem has been stored as a MongoDB database which is populated and queried through a custom Python “EscalatorDB” class. This proof involves local and global computations with 6560 + 104 = 6664 positive definite quadratic forms in 4 variables, called (basic and auxiliary) “escalator forms”. The logfiles for these global computations are very large, and are available for download from here. My Python EscalatorDB source code can be downloaded from HERE, and the MongoDB database file is available HERE.
Interesting Pictures:
Here are some interesting scatter plots showing some unexpected correlations in the escalator form datasets:
- Cusp dimension vs. Maximal Galois degree: The top line correlation is expected (since many of these spaces will have only one Galois orbit of modular forms), but the second smaller line is a little surprising.
- Maximal Galois degree vs. Cusp constant: This plot is very surprising, since it shows that the cusp constants seem to be uniformly small when the maximal Galois degree gets large. Since the maximal Galois degree is a good measure of the computational difficulty of the problem, this plot suggests that computationally hard problems should have easy answers!
- Level vs. Cusp constant: This shows how the cusp constants are distributed as we vary the level. It shows a much weaker correlation than the previous plot, so the size of the level doesn’t really predict too much about the eventual answers, though the answers are usually small.
- Computational time vs. Cusp constant: This plot shows in a very striking way that escalator forms whose theta series are computationally hard to decompose into normalized eigenforms have a uniformly bounded cusp constant (which is < 50).
