I asked myself which primes $p$ can be written as $p=x^3+y^3+z^3-(x^2z+y^2x+z^2y)$ with $x,y,z \in \mathbb{N}$. But for $p \neq 2$ we have the solution $x=y=\frac{p-1}{2}$ and $z=\frac{p+1}{2}$. So the only prime for which I can not find a solution is $p=2$. But I can not prove that there is not a solution. Any ideas?

  • $\begingroup$ Just out of curiousity: What is the matrix $M$? $\endgroup$ – Jose Arnaldo Bebita-Dris Jul 29 '17 at 15:12
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    $\begingroup$ @JoseArnaldoBebitaDris: I edited the question. $\endgroup$ – orgesleka Jul 29 '17 at 15:15
  • $\begingroup$ Look at the equation modulo 4. $\endgroup$ – user940 Jul 29 '17 at 16:57

EDIT: You're right stackExchangeUser; my proof doesn't work. With a similar tack, we can still salvage this: \begin{align*} &x^3 + y^3 + z^3 - (x^2 z + y^2 x + z^2 y) \\ = ~ &(x + y + z)^3 - 4(x^2 z + y^2 x + z^2 y) - 3(x^2y + y^2 z + z^2 x) - 6xyz \end{align*} So, we are solving, $$(x + y + z)^3 = 2 + 4(x^2 z + y^2 x + z^2 y) + 3(x^2y + y^2 z + z^2 x) + 6xyz$$ Suppose first the left side is divisible by $2$. Again, at least one of $x, y, z$ must be even. If all of them are even, we see that the left side is $0$ mod $8$, but the right side is $2$ mod $8$. Thus, exactly one must be even. But then, the $3(x^2y + y^2 z + z^2 x)$ term is odd, which makes the right hand side odd, and we get a contradiction again.

Thus, the left side is odd. Similarly, either all of $x, y, z$ are odd, or exactly one is. If exactly one of them is odd, then the right hand side is even, hence all $x, y, z$ are odd.

Finally, considering the original formulation, and the fact that $x^2 \equiv 1$ mod $8$ for all odd $x$, we get,

$$x^3 + y^3 + z^3 - (x^2 z + y^2 x + z^2 y) \equiv x + y + z - (z + x + y) \equiv 0$$

mod $8$, which cannot equal $2$.

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    $\begingroup$ sage: expand($(x+y+z)^3-4*(x^2*z+y^2*x+z^2*y)-6*x*y*z$) $x^3 + 3*x^2*y - x*y^2 + y^3 - x^2*z + 3*y^2*z + 3*x*z^2 - y*z^2 + z^3$ . I do not understand your first equality. SAGE tells me something different. $\endgroup$ – orgesleka Jul 29 '17 at 15:21

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