Timeline for Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2014 at 1:03 | comment | added | GH from MO | We are now at $m/\ell\approx 0.485$ which is smaller than all previous ratios, including the one for $(\ell,m)=(5,3)$. | |
| Aug 25, 2014 at 20:40 | comment | added | GH from MO | I have now found a pair with $m/\ell\approx 0.766$! See my added section. | |
| Aug 25, 2014 at 15:48 | comment | added | GH from MO | Sure, that was a typo, thanks for spotting it! | |
| Aug 25, 2014 at 15:47 | comment | added | Lucia | @GHfromMO: Your values seem to give $C$ more like $0.04$ rather than $0.4$? (I mean the $C$ in the $C \log x$ of my answer.) This is why I thought there may be a typo there. | |
| Aug 25, 2014 at 15:41 | comment | added | GH from MO | I think the value for $C$ is ok. The initial factors $1/6$, $1/2$, $2/3$ should be clear, while the product over $p>3$ is very close to $1$. The residue of the Dedekind zeta function is $2\pi h_K R_K/|D_K|^{1/2}$ with the usual notation. We have $h_K=1$, $R_K\approx 1.347$, $D_K=-108$. On the other hand, I was wrong saying that any interval of additive length $\approx\exp(1/C)$ should have an $\ell$. Instead, any interval of multiplicative length $\approx\exp(1/C)$ should have one. This means that realistically the first counterexample can be as large as $10^{20}$! | |
| Aug 25, 2014 at 15:29 | comment | added | Lucia | @GHfromMO: That's very nice! If we think of a Poisson process, then maybe one expects to find no counterexample around size $10^{10}$ with probability $1/e$, and finding no counterexample of size $10^{20}$ with probability $1/e^2$ etc. Also, I'm not completely sure of the numerical value you give for $C$ -- maybe a typo somewhere? (If $C$ is really as big as $0.4$ wouldn't you expect counterexamples sooner?) | |
| Aug 25, 2014 at 14:42 | comment | added | GH from MO | I made your heuristic a bit more specific, and performed more extensive experiments. See my added section | |
| Aug 25, 2014 at 0:23 | history | edited | Lucia | CC BY-SA 3.0 |
added 926 characters in body
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| Aug 24, 2014 at 18:09 | history | edited | GH from MO | CC BY-SA 3.0 |
fixed some typos
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| Aug 24, 2014 at 18:04 | history | edited | GH from MO | CC BY-SA 3.0 |
I fixed some typos.
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| Aug 24, 2014 at 17:48 | history | answered | Lucia | CC BY-SA 3.0 |