<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] If $a^2+b^2=c^2,$ with $a,b,c$ positive, you'd have: $c>a,b>0$ and thus $ca^2>a^3$ and $cb^2>b^3.$ So you get:
$$c^3=c\cdot c^2=c(a^2+b^2)=ca^2+cb^2>a^3+b^3.$$
So $c^3>a^3+b^3.$

You can prove more generally for any triple $(a,b,c)$ of positive reals, there is at most one positive $n$ such that $a^n+b^n=c^n.$ If $a^n+b^n=c^n$ then $c>a,b>0$ and for $m>n$ you have:
$$\begin{align}c^m&=c^{m-n}\cdot c^n
\\&=c^{m-n}(a^n+b^n)\\&=c^{m-n}a^n+c^{m-n}b^n\\
&>a^{m-n}a^n+b^{m-n}b^n\\
&=a^m+b^m.\end{align}$$
</s>