Timeline for Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?
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| Mar 17, 2022 at 15:27 | comment | added | dan_fulea | I am thinking that structural problems are of interest. So $\Bbb Z$ is a ring, the ring of integers. The natural numbers are not building a ring. Diophantine equations are usually involving integers. See the first proposition in en.wikipedia.org/wiki/Diophantine_equation. Methods like going $p$-adically are working best with integers. Well, positivity is maybe related to the $\infty$-place, so i was saying that this condition is "somehow unnatural". This is to make clear what i was writing above. Else, above there are still questions, some potential answerer may be helped to start. | |
| Mar 17, 2022 at 13:59 | comment | added | Zhi-Wei Sun | If you think that only homogenous diophntine equations with integer (not natural number) solutions are interesting, I have nothing else to say. | |
| Mar 14, 2022 at 11:12 | comment | added | dan_fulea | The motivation is in a deep (structural) contrast with the question. In the motivation we have a homogeneous quadratic form, $q(\ (x,y,z)\ ):=x^2+y^2+z^2$, and for such a form it is a natural question to ask for the numbers representing it. In the question, there is a "mixed" monomial part, we have terms of degree two, and one monomial of degree three. (So we do not even have a trilinear form to be extracted.) Also, the restriction to natural numbers (versus integers) is somehow unnatural. Why is this question of interest? If it is / would be solved, which is the benefit? | |
| Mar 13, 2022 at 22:29 | comment | added | Zhi-Wei Sun | I have extended my check. The question has a positive answer for $n\le 10^6$. | |
| Mar 13, 2022 at 7:20 | history | asked | Zhi-Wei Sun | CC BY-SA 4.0 |