Generalizing the 290 theorem.

I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic form represents all positive integers it suffices to check that it represents a specific set of 29 integers (the largest of which is 290). Other similar theorems exist too about representing all primes or all odd numbers.

I just read the paper by Bhargava and Hanke and found the proof quite elegant. I got thinking about whether it would generalize to other situations.

Has anyone has been able to extend the results to other settings? Maybe people have been able to prove similar things over other rings (such as rings of integers of number fields) or maybe people are still considering staying in the integer case and considering representing other sets of integers or considering higher degree forms?

Quite a bit of information is available as pdfs at my page TERNARY with what I hope are obvious names.

You want to look at Rouse on all odd numbers, ROUSE. Also Representation by ternary quadratic forms by OLIVER. In both cases some ineffective bounds are used, so a GRH is invoked that implies the suspected conclusions. This gives about the best conclusion to my paper with Kaplansky and Schiemann that I have any right to expect.

Hanke certainly thought that almost anything could be extended to integer rings of some number fields, and intended to find all class number one genera. This was an ambitious project, as it would require dimension up to 26. A student of Gabriele Nebe, named David Lorch, has found all positive class number one forms over $\mathbb Z,$ see LORCH.

I do not believe I know of any big-list papers on universality over number rings. There are some related approaches by Pete L. Cark of MO and MSE, see item 15 at CLARK. In this case, there was surely some influence by Hanke, who was at Georgia for some years.

Here is my webpage with information about the 290-Theorem. It has links to the preprint, all escalator form datafiles and computer code. In Spring 2014 I had a student (Kate Thompson) graduate from UGA after learning about the analytic techniques involved in doing similar computations for totally positive definite $\mathcal{O}_F$-valued quadratic forms over $\mathcal{O}_F := \mathbb{Z}\left[\frac{1 + \sqrt{5}}{2}\right]$. If you're interested in learning more about the theory behind this theorem, here is a video of the talk I gave at Rutgers last year about it.

It's certainly possible to prove a version of the 290-Theorem to establish universality of forms over the ring of integers $\mathcal{O}_F$ of an arbitrary totally real number field $F$ (modulo some conditions about no ternary escalator forms appearing), but it is a highly non-trivial task to create a general code base to prove what numbers are represented by an arbitrary totally definite $\mathcal{O}_F$-valued quadratic form in 4 variables. This is what was done over $\mathbb{Z}$ for the 290-Theorem, and then was used on 6664 quadratic forms to establish the result. The point is that doing it without a computer is not feasible, but writing a program to accurately perform this task is very difficult and very time consuming (and may adversely affect someone's career unless they have tenure). To give you a sense of it -- this project took me about 4 years of focused work to initially complete the algorithm development and coding/debugging, and several more years to recheck the computations to my satisfaction.

I'm sure that the paper will appear as more than a preprint at some point, and it's certainly not something that I've forgotten about. =)