Timeline for Extending rational Diophantine triples to sextuples
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 13, 2019 at 15:38 | answer | added | duje | timeline score: 2 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 22, 2016 at 4:08 | answer | added | Tito Piezas III | timeline score: 5 | |
| Mar 18, 2016 at 20:18 | answer | added | Tito Piezas III | timeline score: 5 | |
| Mar 16, 2016 at 21:45 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Improved formatting.
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| Mar 13, 2016 at 21:34 | comment | added | Tito Piezas III | @duje: By the way, I've analyzed Gibbs' 45 sextuples and found 12 that belong to a family distinct from yours. (They do not contain triples that obey $(4)$.) I believe a different elliptic curve is involved. I'll post the family here when I've studied it more. | |
| Mar 13, 2016 at 21:20 | comment | added | Tito Piezas III | @duje: Oh, I just recognized your name. :) By the way, in your paper, Lemma 1 can be aesthetically expressed as, $$\big(-3 - (a + b + c)^2 + 2 (a^2 + b^2 + c^2)\big)\,(1 + a^2 b^2 c^2) = (a + b + c + 3 a b c)^2$$ Since the second factor of the $LHS$ is a square, that implies the first one is as well and implies these special triples obey $(4)$ above. | |
| Mar 13, 2016 at 21:14 | comment | added | duje | The formula $v=\frac{t^2-1}{2t}$ comes from the condition $1+v^2$ is a square. And formula for $t$ in terms of $T$ comes from the condition $t^2+8$ is a square, which gives an additional point on the elliptic curve, so that it has rank >=2. There are some other similar conditions which also lead to rank >=2. We plan to summarize these constructions in a paper soon. | |
| Mar 13, 2016 at 21:07 | comment | added | Tito Piezas III | @duje: The elliptic curve you cite below was specialized from $v = \frac{t^2-1}{2t}$. Do you think we can find another $v$ that may also yield simple families? | |
| Mar 13, 2016 at 21:05 | comment | added | duje | It might be interesting to mention that elliptic curves y^2=(ax+1)(bx+1)(cx+1) of this shape have torsion group Z/2Z x Z/6Z over Q. In fact, the most of known (and some new) record rank curves with this torsion (over Q and over Q(t)) can be obtained from the triple (5) by specializations of the parameter t (this is joint work in progress with J. P. Peral). | |
| Mar 13, 2016 at 20:51 | vote | accept | Tito Piezas III | ||
| Mar 13, 2016 at 20:31 | answer | added | duje | timeline score: 7 | |
| Mar 13, 2016 at 20:14 | comment | added | duje | In terminology of mentioned paper formula (5) corresponds to the point 2R, while formula (6) corresponds to the points 3R. Other multiples of R produce less esthetical formulas. | |
| Mar 13, 2016 at 20:00 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Phrasing.
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| Mar 13, 2016 at 19:49 | history | asked | Tito Piezas III | CC BY-SA 3.0 |