Timeline for Density of multi-grade solutions to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$ for $k = 5$ or $6$?
Current License: CC BY-SA 3.0
19 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 4, 2015 at 16:00 | answer | added | Wolfgang | timeline score: 3 | |
| Jan 4, 2015 at 6:03 | vote | accept | Tito Piezas III | ||
| Jan 3, 2015 at 18:24 | comment | added | Tito Piezas III | @Wolfgang: Yes, so the non-empty intersections in Moore's database (which are positive term) are only $X_5 \cap X_1$ and $X_6 \cap X_2$. | |
| Jan 3, 2015 at 17:51 | comment | added | Wolfgang | Interesting. And similar for $k=3,4$? | |
| Jan 3, 2015 at 17:47 | comment | added | Tito Piezas III | @Wolfgang: I checked Table 2 ($k=6$, with $405$ solutions). Not a single one is valid for $k=1$ (if terms are positive). This is in stark contrast with Table 1 ($k=5$) wherein the first $168$ solutions have $63.7\text{%}$ valid for $k=1$ as well. | |
| Jan 3, 2015 at 17:39 | comment | added | Tito Piezas III | @DanielLoughran: I'm basing the counts purely on Moore's database so you are correct, I'm not counting a) "trivial solutions" ($x_i=y_i$); and also not counting b) "non-primitives" (those with all terms having a common factor). | |
| Jan 3, 2015 at 15:22 | answer | added | Daniel Loughran | timeline score: 9 | |
| Jan 3, 2015 at 11:10 | comment | added | Wolfgang | Have you checked whether some of the non trivial solutions are also valid for other $k$'s between 1 and 6 (or maybe even a bit bigger)? It would be even more puzzling if there is a special 'link' between 1,5 and between 2,6. | |
| Jan 3, 2015 at 10:36 | comment | added | Daniel Loughran | You have not said so, however it appears that you are not counting "trivial solutions", e.g. when $x_i = y_i$ for all $i$. Could you please clarify which solutions you are counting? Are there are other solutions which you not count? | |
| Jan 3, 2015 at 7:13 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Wording.
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| Jan 3, 2015 at 7:03 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Added clarification.
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| Jan 3, 2015 at 6:47 | comment | added | Tito Piezas III | @DanielLoughran: Ok. For $B$, as an example, inspecting the first $50$ solutions of Moore's database for $k=6$, then $80\text{%}$ of them are actually valid for both $k=2,6$. If we inspect the first $100$ solns, then $85\text{%}$ of them are so valid. And so on. | |
| Jan 3, 2015 at 6:35 | comment | added | Daniel Loughran | Can you please clarify the definitions of $A$ and $B$? It is not clear to me at the moment how these are defined. | |
| Jan 2, 2015 at 23:23 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Simplified table. Added note.
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| Jan 2, 2015 at 21:54 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
MO link.
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| Jan 2, 2015 at 21:36 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Clarify
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| Jan 2, 2015 at 21:28 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Grammar
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| Jan 2, 2015 at 20:56 | history | asked | Tito Piezas III | CC BY-SA 3.0 |