Timeline for Integral points on a particular family of curves
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 2, 2015 at 12:05 | answer | added | castor | timeline score: 1 | |
| Jul 22, 2015 at 6:13 | comment | added | joro | I edited my answer for the case $n=4$ giving map to the original curve. | |
| Jul 21, 2015 at 20:44 | answer | added | Jeremy Rouse | timeline score: 5 | |
| Jul 21, 2015 at 20:28 | comment | added | Jeremy Rouse | For a hyperelliptic curve $y^{2} = f(x)$, where $f(x)$ has even degree, there are two points at infinity on the non-singular model. In addition, there are the points $(-5/2,3/2)$ and $(-5/2,-3/2)$. This explains the four points other than those with $y = 0$. | |
| Jul 21, 2015 at 18:47 | comment | added | Vesselin Dimitrov | Well, one of them is just the point $[0:1:0]$ at infinity, which is the neutral element. So there are three more. | |
| Jul 21, 2015 at 18:45 | comment | added | Igor Rivin | @VesselinDimitrov Yes, exactly. But what about those four other mysterious points. | |
| Jul 21, 2015 at 18:44 | comment | added | Vesselin Dimitrov | I do not know if I can read the conclusion of no other integral solutions without actually computing all seven non-trivial points of the transformed equation, as well as the change of coordinates map. In any case, we know that four of these seven non-identity rational points were given as $y = 0$ and $x \in \{-1,\ldots,-4\}$ in the original coordinates. | |
| Jul 21, 2015 at 18:24 | comment | added | Igor Rivin | @VesselinDimitrov Oops, forgot that $4$ was a square :), thanks! But still, can you parse what the sage result actually means? | |
| Jul 21, 2015 at 18:19 | comment | added | Vesselin Dimitrov | Regarding the addendum: For $n = 4$, there are no non-trivial ($y \neq 0$) integer solutions, by the Erdos-Selfridge result that I quoted. | |
| Jul 21, 2015 at 18:14 | history | edited | Igor Rivin | CC BY-SA 3.0 |
added sage results, added tag
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| Jul 21, 2015 at 7:04 | answer | added | Vesselin Dimitrov | timeline score: 7 | |
| Jul 21, 2015 at 6:33 | answer | added | joro | timeline score: 2 | |
| Jul 21, 2015 at 3:45 | history | asked | Igor Rivin | CC BY-SA 3.0 |