<s>[INST] Does any (right) triangle exist such that $a^3+b^3=c^3$? 
Does any right triangle exist such that $a^3+b^3=c^3$? Does any triangle exist such that $a^3+b^3=c^3$?

I'm stuck on this problem; I tried applying the Pythagorean theorem in three dimensions, but in vain. Any tips? 

 [/INST] We need $a\ne0$ and $b\ne0$. If it holds, then we have
$(a^3+b^3)^2=c^6=(a^2+b^2)^3$
$a^6+2a^3b^3+b^6=a^6+3a^4b^2+3a^2b^4+b^6$
$2ab=3a^2+3b^2$
$(a-b)^2+2a^2+2b^2=0$
$a=b=0$
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