<s>[INST] When $\frac {a^3-b^3}{a^2-b^2}$ is an integer? If $\frac {a^3-b^3}{a^2-b^2}$ is an integer, then supposing $a-b \ne 0$ we have that also$\frac {a^2+ab+b^2}{a+b}$ is an integer. 

For which $a, b\in\mathbb Z$, the fraction $\frac {a^2+ab+b^2}{a+b}$ is an integer?

 [/INST] Equation, $(a^2+ab+b^2)=p(a+b)$
where "p' is integral. 
while solution given by "Jack D'Aurizio" is nice & since
'OP' needs $(a,b)$ to be integer's there is a 
fraction $(k^2/d)$ to be taken care of in his solution.
If instead we take, $(a,b)=[d(k+1),dk(k+1)]$ then we get:
$p=d(k^2+k+1)$
For, $(d,k)=(5,2)$ we get: 
$(a,b,p)=(15,30,35)$
</s>