<s>[INST] Squares of the form $x^2+y^2+xy$ How can I find all $(a,b,c) \in \mathbb{Z}^3$ such that $a^2+b^2+ab$, $a^2+c^2+ac$ and $b^2+c^2+bc$ are squares ?
Thanks !
 [/INST] I have shown here that:
All coprime triples $(a,b,c)$ so that $a^2 + ab + b^2 = c^2$ can be
enumerated, without duplication, by taking two positive integers
$m \ge n$, where $3$ does not divide $n$, and either $mn$ is odd and
$\gcd(m,n) = 1$, or $8$ divides $mn$ and $\gcd(m,n) = 2$, and by setting
$$
\begin{align}
a&=mn\tag{1a}\\[9pt]
b&=\frac{(3m+n)(m-n)}{4}\tag{1b}\\[9pt]
c&=\frac{3m^2+n^2}{4}\tag{1c}
\end{align}
$$
</s>