<s>[INST] Determine all pairs of positive integers $(a,b)$ such that $a^2+b^2+ab$ is a perfect square. Consideration via mod $4$ shows that $a,b=0$ mod $4$ or one of between $a,b$ is $=1$ mod $4$ while the other is $=0 $ mod $4$.
Considering $(a+b)^2,a^2+an+b^2,(a+b-1)^2$ as we can deduce that $a,b>2$.
 [/INST] Primitive solutions to $a^2+ab+b^2=c^2.$
Let $x=a/c, y=b/c.$
$x^2+xy+y^2=1\tag{1}$
Substitute  $x = t , y=1+kt$  to equation $(1)$, then we get  $$t = -\frac{(1+2k)}{(1+k+k^2)}$$
Thus, we get a parametric solution.  
$(x,y,z) = (-\frac{(1+2k)}{(1+k+k^2)}, -\frac{(k-1)(k+1)}{(1+k+k^2)})$ 
Hence $(a,b,c)=(1+2k, k^2-1, 1+k+k^2)$
$k$ is arbitrary. 
Further, take $k=\frac{\alpha}{\beta}$ then we get
$(a,b,c)=(\beta^2+2\alpha\beta,\alpha^2-\beta^2
,{\beta}^{2}+\alpha\,\beta+{\alpha}^{2})$
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