<s>[INST] Finding integer solutions to $ \frac{a^3+b^3}{a^3+c^3} = \frac{a+b}{a+c} $ I was browsing through facebook and came across this image: 
I was wondering if we can find more examples where this happens?
I guess this reduces to finding integer solutions for the equation 
$$ \frac{a^3+b^3}{a^3+c^3} = \frac{a+b}{a+c} $$ for integers a,b,c
Or can we even further extend to when they are all distinct that is finding solutions to 
$$ \frac{a^3+b^3}{c^3+d^3} = \frac{a+b}{c+d} $$ for integers a,b,c
I don't really have that much knowledge in the number theory area so I have come here
 [/INST] Let $S$ be the set of allowed values of $a$, $b$, and $c$ ($S=\mathbb{Z}$ in this OP's setting, but $S$ can be something else like $\mathbb{Q}$, $\mathbb{Q}_{>0}$, $\mathbb{R}$, or even $\mathbb{F}_p$, where $p$ is a prime natural number).  If $b=-a$, then $c$ can be any number not equal to $-a$.  That is, $(a,b,c)=(a,-a,c)$ with $c\neq -a$ is always a solution.  From now on, we assume that $b\neq-a$.  
Then, as invidid found,  $$\frac{a^2-ab+b^2}{a^2-ac+c^2}=\left(\frac{a^3+b^3}{a^3+c^3}\right)\left(\frac{a+b}{a+c}\right)^{-1}=1\,.$$
Hence, $a^2-ab+b^2=a^2-ac+c^2$, or
$$(b-c)(b+c-a)=0\,.$$
That is, $b=c$ or $a=b+c$.  
This concludes that all solutions $(a,b,c)\in S^3$ takes the form $(a,-a,c)$ with $c\neq -a$, $(a,b,b)$ with $b\neq -a$, and $(b+c,b,c)$ with $c\neq -\frac{b}{2}$.  You can check that these indeed are solutions.  The two solutions $(a,b,c)=(5,2,3)$ and $(a,b,c)=(579,123, 456)$ that you found are of the form $(a,b,c)=(b+c,b,c)$.
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