<s>[INST] Finding integers $a,b$ and $c$ such that $a^3+b^3 = c^3$ We are learning about the the Pythagorean Theorem in class. It says that $a^2+b^2 = c^2$. My homework problem says the following: 

Find integers $a,b$ and $c$  such that $a^3+b^3 = c^3$. 

How do I solve this equation? 
I've been starting with $(3,4,5), (4,4,5)$ etc. Basically I am starting Pythogrean triples.
 [/INST] If $a=0,b^2=c^2\implies b=\pm c$  and $b^3=c^3\implies b=c$
$\implies b=c$
So, $(0,b,b)$ is a solution
Similarly, $(b,0,b)$ is a solution
If $c\ne0, \left(\frac ac\right)^2+\left(\frac bc\right)^2=1$
$\implies \frac ac<1\implies \left(\frac ac\right)^2>\left(\frac ac\right)^3$
Similarly, $\left(\frac bc\right)^2>\left(\frac bc\right)^3$
$\implies \left(\frac ac\right)^3+\left(\frac bc\right)^3<1$
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