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Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$.

Motivated by this, here I pose the following question.

Question. Whether for any $n\in\mathbb N$ we can write $12n+5$ as $2x^2+5y^2+9z^2+xyz$ with $x,y,z\in\mathbb N$?

I have verified that for each $n=0,\ldots,3\times10^5$ we can write $12n+5$ as $2x^2+5y^2+9z^2+xyz$ with $x,y,z\in\mathbb N$. For example, $12\times21030+5=252365$ has a unique desired representation: $$252365 = 2\times32^2 + 5\times126^2+9\times39^2+32\times126\times39.$$

I guess that the question has a positive answer. Any ideas to approach this?

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  • $\begingroup$ I have extended my check. The question has a positive answer for $n\le 10^6$. $\endgroup$
    – Zhi-Wei Sun
    Commented Mar 13, 2022 at 22:29
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    $\begingroup$ 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? $\endgroup$
    – dan_fulea
    Commented Mar 14, 2022 at 11:12
  • $\begingroup$ If you think that only homogenous diophntine equations with integer (not natural number) solutions are interesting, I have nothing else to say. $\endgroup$
    – Zhi-Wei Sun
    Commented Mar 17, 2022 at 13:59
  • $\begingroup$ 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. $\endgroup$
    – dan_fulea
    Commented Mar 17, 2022 at 15:27

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