Timeline for Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 26 at 22:08 | vote | accept | Dima Pasechnik | ||
| Nov 26 at 1:16 | answer | added | GH from MO | timeline score: 5 | |
| Nov 26 at 0:15 | history | edited | LSpice | CC BY-SA 4.0 |
Link to comment
|
| Nov 25 at 20:50 | comment | added | Dima Pasechnik | x and y are allowed to be rationals. | |
| Nov 25 at 19:41 | comment | added | Peter Mueller | Are $x$ and $y$ supposed to be integers, or allowed to be rationals? | |
| Nov 25 at 19:13 | comment | added | Aleksei Kulikov | Most likely the answer is no, see this MO question mathoverflow.net/questions/373900/… (it deals with the case of integer coefficients, but I would assume it is not much different for rational). | |
| Nov 25 at 19:12 | history | edited | Ofir Gorodetsky | CC BY-SA 4.0 |
deleted 14 characters in body
|
| Nov 25 at 19:08 | history | edited | Dima Pasechnik | CC BY-SA 4.0 |
rectify
|
| Nov 25 at 19:08 | comment | added | Ofir Gorodetsky | Yes, things are different if you restrict $n$ to be a prime. For each $a,b\ge 0$, the representable set has positive density within the primes (e.g. for $a=b=1$, this is due to Dirichlet). The precise density is related to class field theory, see David A. Cox's book "Primes of the form x² + ny²". So it is likely that in this case finitely many $(a_k,b_k)$-s are sufficient -- but I don't think this was worked out (a version of this has been asked as Problem 4 here: math.purdue.edu/~sahay5/frg-problem-session.pdf ). | |
| Nov 25 at 19:05 | comment | added | Dima Pasechnik | Would the density be different for $n$ prime? | |
| Nov 25 at 18:54 | comment | added | Ofir Gorodetsky | For fixed $a,b \in \mathbb{Q}_{\ge 0}$ the representable set has density $0$ (for $a=b=1$ this is due to Landau; for general $a,b\ge 0$ this is due to Bernays, at least for integers $a,b$). So a finite set of $(a_k,b_k)$-s is not sufficient. However, see Green and Soundararajan's preprint "Covering integers by x^2+dy^2" for a positive result: arxiv.org/abs/2401.04817 | |
| Nov 25 at 18:49 | history | asked | Dima Pasechnik | CC BY-SA 4.0 |