Timeline for Separation of lattice points on the Mordell elliptic curve
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 8, 2014 at 12:52 | comment | added | Joe Silverman | Ryan, why don't you send me an email so that we can discuss this offline. You can find my email address on the webpage that's linked on my MathOverflow home page. -- JS | |
| Sep 8, 2014 at 4:01 | comment | added | Ryan D'Mello | @JoeSilverman As promised, here is my update. I am able to prove that x2 - x1 is at least roughly (x1)^(1/8). Actually, I believe that I can improve on this result as follows: for any e > 0, there is a constant c(e) (depending only on e) such that for all x(n) > c(e), x(n + 1) – x(n) is roughly at least x(n)^(1/(6 + e)). Hence, the separation gap between the x(n) gets closer to x(n)^(1/6) as x(n) gets large. I can also calculate c(e) explicitly in terms of e. I would appreciate your opinion on how useful and/or interesting you think this result is. | |
| Aug 28, 2014 at 23:37 | comment | added | Ryan D'Mello | @JoeSilverman I'm very sorry that the words I used did not come out sounding like what I meant to say (it was late at night!). My apologies. I did understand exactly what you had meant. Regarding your suggestion, I tried similar approaches months ago but kept running into a dead-end. There may be some clever way to make it work. My approach resulted from a revelation while inspecting output from a JAVA program I wrote to explore x^3 - y^2 differences (for several billion x). Not sure if this is the right forum to share some of that background, but I can email you offline if you are interested. | |
| Aug 27, 2014 at 14:47 | comment | added | Joe Silverman | @RyanD'Mello Actually, if you assume that $(x,y)$ is a solution with $x$ reasonably large and $|x^3-y^2|\asymp\sqrt{|x|}$, and if you let $(x+\ell,y+k)$ be the next largest solution, then approximating $(x+\ell)^3\approx x^3+3\ell x^2$ and $(y+k)^2\approx y^2+2ky$ gives $k$ in terms of $\ell$ (more or less), and then looking at the next largest terms may well give a lower bound for $\ell$ in terms of $x$. I'll let you work out the details (and maybe this doesn't work at all). | |
| Aug 27, 2014 at 14:42 | comment | added | Joe Silverman | @RyanD'Mello Please don't misquote me. I didn't say that there is no simple answer using currently known (or even possibly quite elementary) methods. I simply said that at a quick glance, the results/methods in my paper don't seem to yield anything, and that offhand I don't recall seeing this problem addressed in the literature (which is a much weaker statement than saying that it hasn't been considered before!). | |
| Aug 27, 2014 at 3:07 | comment | added | Ryan D'Mello | @JoeSilverman I agree. I've been working on finalizing a proof and am more motivated now after learning that there is no simple answer to this question using currently known methods. I hope to have an update in about a week or two. Thanks again for your valuable input and opinions. | |
| Aug 26, 2014 at 11:00 | comment | added | Joe Silverman | @RyanD'Mello Offhand I don't know what lower bound on $x_2-x_1$ can be regarded as trivial and what would be significant. The best way to figure that out is to try to prove something, which I'll let you do, rather than my working more on the problem. | |
| Aug 26, 2014 at 6:24 | comment | added | duje | @RyanD'Mello I agree with you that my result could provide only an upper bound for a gap. | |
| Aug 26, 2014 at 4:21 | comment | added | Ryan D'Mello | @Joe Silverman Sorry, in my previous comment, I should have had x2 - x1 > log(x1) instead of x1 - x2 > log(x1) | |
| Aug 26, 2014 at 4:16 | comment | added | Ryan D'Mello | @duje Thanks for your suggestions. Actually, I did run across your paper a month ago while I was exploring this question, but was not sure how it would provide a LOWER bound for a gap. It seems like this would provide an upper bound for a gap, or am I missing something?? For example, do you see your result answering either of the questions I posed in the comment just above to Joe Silverman (regarding separation by log(x1) or even 2)? | |
| Aug 26, 2014 at 4:07 | comment | added | Ryan D'Mello | @Joe Silverman Thanks for providing your opinion that this is an interesting question and one that is not easily answered. That is precisely the type of professional opinion I needed, so this is very helpful. (Your rank-based bound on the number of integer points is neat.) Suppose something as basic as the following could be proved: If (x1,y1) and (x2,y2) are integer points on x^3 - y^2 = k, k^2 < x1 < x2, then x1 - x2 > log(x1). Would you consider that an interesting and/or useful result? What if log(x1) is replaced by 2? Do you agree that even the proof for 2 is not trivial/obvious? | |
| Aug 25, 2014 at 17:15 | comment | added | duje | Concerning the inequality $0<|x^3-y^2|<x^{1/2+\varepsilon}$, I proved in the paper On Hall's conjecture, Acta Arith. 147 (2011), 397-402. (web.math.pmf.unizg.hr/~duje/pdf/hall3.pdf) that there are $\gg N^{\varepsilon/(5+4\varepsilon)}$ solutions in the range $1\leq x \leq N$. | |
| Aug 25, 2014 at 12:46 | comment | added | Joe Silverman | I'd forgotten about that Danilov article. Anyway, I thought about it for a little bit and don't see an immediate way to get a gap estimate using (local) canonical height estimates. Seems like an interesting problem. My article with an elliptic curve gap principle is: A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60-100. For the curves $E_k:Y^2=X^3+k$ with $k$ 6'th power free, I prove that there is an absolute constant $C$ so that $|E_k(\mathbb{Z})|\le C^{1+rank~E_k(\mathbb{Q})}$`. | |
| Aug 25, 2014 at 5:04 | comment | added | Ryan D'Mello | @Joe Silverman It is known that there are infinitely many x such that there is a y satisfying 0 < |x^3 - y^2| < sqrt(X). In fact, there is what is known as the Danilov-Elkies infinite family, every element of which has this property. This is discussed in Elkies' paper referenced in my original post. (Also see oeis.org/A200216 for some examples of elements in this family.) I am very interested in learning more about your general gap principle for integral points on elliptic curves. What does it state? | |
| Aug 25, 2014 at 4:48 | comment | added | Ryan D'Mello | Joe & Lucia, thanks for the responses. The question I posed is distinct from (but has obvious implications for) lattice point separation on a Mordell elliptic curve C: x^3 - y^2 = k (when the x-coordinates of the lattice points are > k^2). I was aware of the Hall/Davenport connection but, as Lucia pointed out, my question has to do with gaps between successive non-square x which have the property that the distance between x^3 and the square nearest to x^3 is less than sqrt(x). So, there are two distinct but related questions here, with the first having some implications for the second ...... | |
| Aug 24, 2014 at 23:13 | comment | added | Joe Silverman | @Lucia That would be a gap principle sort of statement. I'll have to think about it. I wrote a paper with a general gap principle for integral points on elliptic curves, but I'm not sure if it's relevant here. OTOH, if the specific question is about the $x\in\mathbb{N}$ such that there is a $y$ satisfying $0<|x^3-y^2|<\sqrt{x}$, it's not at all clear (at least to me) that the set of such $x$ is infinite. But maybe one could fix a small $\epsilon$ and take $x$ values admitting a solution to $0<|x^3-y^2|<x^{1/2+\epsilon}$, or use an upper bound of $x^{1/2}(\log x)^k$ for some fixed $k$. | |
| Aug 23, 2014 at 15:39 | comment | added | Lucia | To clarify: your question is not about the difference between $x^3$ and $y^2$ (Hall's conjecture), but rather about the difference between those values of $x$ for which the spacing between $x^3$ and its nearest square is small (below $\sqrt{x}$). Is that correct? | |
| Aug 23, 2014 at 11:39 | answer | added | Joe Silverman | timeline score: 5 | |
| Aug 22, 2014 at 23:44 | review | First posts | |||
| Aug 22, 2014 at 23:56 | |||||
| Aug 22, 2014 at 23:40 | history | asked | Ryan D'Mello | CC BY-SA 3.0 |