Prove there do not exist prime numbers $a$, $b$, and $c$ such that $a^3+b^3=c^3$ Prove there do not exist prime numbers $a$, $b$, and $c$ such that $a^3+b^3=c^3$.
From what I understand this proof requires a proof by contradiction or contrapositive...

$a^3+b^3=(a+b)(a^2-ab+b^2)$ So we have
$(a+b)(a^2-ab+b^2)=c^3$
So $(a+b)=c$ and $(a^2-ab+b^2)=c^2$ 
Hence $(a+b)^2=a^2+2ab+b^2=a^2-ab+b^2$ which is a contradiction since it implies $3ab=0$
