## Task 19/87

Which natural numbers $n$ cannot be represented in the form $n=x^{2}+2 y^{2}+3 z^{2}$ with $x ; y ; z \in N$?
All natural numbers $n$ can be represented in the form $n=8 k \pm r$ with $k \in \mathbb{N}, r \in \{0 ; 1 ; 2 ; 3 ; 4\}$. Therefore, it follows that

$$
n^{2}=64 k^{2} \pm 16 k r+r^{2} \equiv r^{2} \quad(\bmod 16)
$$

with $r^{2} \equiv 0 ; 1 ; 4 ; 9(\bmod 16)$. Now let

$$
x^{2} \equiv p \quad(\bmod 16), \quad 2 y^{2} \equiv q \quad(\bmod 16), \quad 3 z^{2} \equiv s \quad(\bmod 16)
$$

for arbitrary $x ; y ; z \in \mathbb{N}$; then it holds that

$p \equiv 0 ; 1 ; 4 ; 9(\bmod 16), q \equiv 0 ; 2 ; 8(\bmod 16), s \equiv 0 ; 3 ; 11 ; 12(\bmod 16)$.

If one forms the $4 \cdot 3 \cdot 4=48$ possible sums $p+q+s$, one finds that the congruence $p+q+s \equiv 10$ (mod 16) has no solution. Therefore, natural numbers $n=16 k+10, k \in \mathbb{N}$ cannot be represented in the given form.

Thus, not all such numbers are covered, but this was not required in the problem.