Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?

Question

It is well known that each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $2w^2+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Furthermore, $$\{2w^2+x^2+y^2:\ w,x,y\in\mathbb N\}=\mathbb N\setminus\{4^k(16m+14):\ k,m\in\mathbb N\}.$$ Motivated by this, here I pose the following novel question.

Question 1. Can each $n\in\mathbb N$ be written as $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?

I guess that the answer is positive, which has been verified for $n\le 10^6$.

Similarly, I have the following four questions.

Question 2. Are $7$ and $487$ the only natural numbers which cannot be written as $w^2+x^2+y^2+z^2+xyz$ with $w,x,y,z\in\mathbb N$?

Question 3. Is it true that each $n\in\mathbb N$ with $n\not\equiv 3\pmod4$ can be written as $4w^2+x^2+y^2+z^2+xyz$ with $w,x,y,z\in\mathbb N$?

I also believe that the answers to Questions 2 and 3 should be positive, which have been verified for $n\le 10^6$. It is easy to see that $$x^2+y^2+z^2+xyz\not\equiv3\pmod4$$ for any $x,y,z\in\mathbb Z$.

Question 4. Is $23$ the only natural number which cannot be written as $w^2+x^2+y^2+z^2+3xyz$ with $w,x,y,z\in\mathbb N$?

I guess that the answer is positive. I have checked this for natural numbers up to $2\times10^6$.

Question 5. Is it true that each $n\in\mathbb N$ with $n\not\equiv3\pmod4$ can be written as $4w^3+x^2+y^2+z^2+xyz$ with $w,x,y,z\in\mathbb N$? Are $7,\,87$ and $267$ the only natural numbers which cannot be written as $w^3+x^2+y^2+z^2+xyz$ with $w,x,y,z\in\mathbb N$?

It seems that Question 5 should also have a positive answer; I have checked this for natural numbers up to $10^5$.

Any ideas to the above new questions? Your comments are welcome!

Answer 1

The answer to question 1 is yes - the other questions seem to me to be more difficult.

If $n$ is odd, then there are non-negative integers $w$, $x$ and $y$ so that $n = 2w^{2} + x^{2} + y^{2}$. One way to see this is that the class number of this quadratic form $Q_{1} = 2w^{2} + x^{2} + y^{2}$ is $1$, and so every locally represented integer is represented. The only local obstructions are at $2$, and this implies that every integer not of the form $14 \cdot 4^{k} \pmod{4^{k+2}}$ for $k \geq 0$ is represented by $Q_{1}$.

If $n$ is even, then there are non-negative integers $w$, $x$ and $y$ so that $n = 2w^{2} + x^{2} + xy + y^{2} + 1$. The quadratic form $Q_{2} = 2w^{2} + x^{2} + xy + y^{2}$ also has class number $1$, and it represents all integers not of the form $10 \cdot 4^{k} \pmod{4^{k+2}}$ for some $k \geq 0$. In particular, $Q_{2}$ represents every odd number and so every even $n$ can be written in the form $2w^{2} + x^{2} + xy + y^{2} + 1$.

Note that if $m \in \mathbb{N}$ and $m = x^{2} + xy + y^{2}$ for some $x, y \in \mathbb{Z}$, then there are $x'$ and $y'$ in $\mathbb{N}$ so that $m = (x')^{2} + (x')(y') + (y')^{2}$. This can be seen by applying one of the six automorphisms of the quadratic form $x^{2}+xy+y^{2}$.