Timeline for What is the roadblock in the discovery of new taxicab numbers?
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11 events
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| Aug 22, 2020 at 15:17 | comment | added | Joe Silverman | @KConrad If you allow iimprimitive solutions, i.e., with $\gcd(x,y)>1$, then you can get as many solutions as you want by clearing denominators from a bunch of rational solutions. The original idea of doing this is due to Chowla. I quantified it to some extent in the paper: Integer points on curves of genus 1, J. London Math. Soc. 28 (1983), 1-7. OTOH, if we restrict to primitive solutions, then maybe we can get a uniform bound for the number of solutions in terms of the rank, even if $m$ is not cube-free. The key is to exploit the canonical height lower bound. | |
| Aug 22, 2020 at 14:59 | comment | added | KConrad | @JoeSilverman do you mean there would be an absolute upper bound on the number of primitive integral solutions $(x,y)$ to $x^3 + y^3 = m$ only for cubefree $m$, or is there a way to bootstrap that to an absolute upper bound allowing general $m$? Your papers on the topic focus on cubefree $m$. | |
| Aug 22, 2020 at 1:17 | history | edited | Glycerius | CC BY-SA 4.0 |
fixed typo
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| Aug 21, 2020 at 18:18 | vote | accept | Glycerius | ||
| Aug 21, 2020 at 0:41 | history | became hot network question | |||
| Aug 20, 2020 at 17:39 | comment | added | Joe Silverman | @AntoineLabelle One of the papers with rank bound heuristics is A heuristic for boundedness of ranks of elliptic curves Jennifer Park, Bjorn Poonen, John Voight, Melanie Matchett Wood arxiv.org/abs/1602.01431 If they're correct, then that old theorem of mine that Stanley mentioned would imply that there is an absolute bound for the number of primitive solutions to $x^3+y^3=m$, where primitive means $\gcd(x,y)=1$. Note that the taxicab numbers do not impose this gcd restriction. | |
| Aug 20, 2020 at 17:38 | answer | added | Joe Silverman | timeline score: 21 | |
| Aug 20, 2020 at 17:31 | comment | added | Antoine Labelle | @StanleyYaoXiao Interesting comment, I find it very surprising that the ranks might be bounded. What are these heuristics more precisely, do you have some reference? | |
| Aug 20, 2020 at 17:14 | comment | added | Stanley Yao Xiao | One reason might be the following: by a theorem of Silverman, if there exists a binary cubic form $F$ with integer coefficients and an infinite sequence $\{h_n\}$ such that the number of primitive solutions to the Thue equation $F(x,y) = h_n$ increases as a function of $n$, then there exist elliptic curves over $\mathbb{Q}$ of arbitrarily large rank. This was widely believed until relatively recently, when heuristics due to a large number of authors indicate that rank of elliptic curves over $\mathbb{Q}$ might in fact be bounded. | |
| Aug 20, 2020 at 16:46 | review | First posts | |||
| Aug 20, 2020 at 16:56 | |||||
| Aug 20, 2020 at 16:40 | history | asked | Glycerius | CC BY-SA 4.0 |