On Solution of FLT for exponent 3

Question

There is a simple proof that $a^3$ + $b^3$ + $c^3 = 0$ has no solutions in integers, is this really the same thing for the actual case of $a^3$ + $b^3 = c^3$, what is the connection between the two equations, how does a no-solution for the first form translates into a no-solution for the second form?

Answer 1

$$a^3 + b^3 = c^3 \iff a^3 + b^3 - c^3 = 0$$ $$\iff a^3 + b^3 + (-c)^3 =0 $$ so (a,b,c) is a solution of the second equation if and only if $(a,b,-c)$ is a solution of the second one, and this gives a bijection between solutions to the two equations over any ring.

Answer 2

I assume for both equations the assumption is that $a, b, c \neq 0$, otherwise both equations have solutions. Alex's solution is right: $(a, b, c)$ satisfies the first equation iff $(a, b, -c)$ satisfies the second. I just want to remark that FLT asserts that there are no solutions to $a^3 + b^3 = c^3$ where $a, b, c \in \mathbb{N}$. However, this is equivalent to asserting that there are no solutions to $a^3 + b^3 = c^3$ where $a, b, c \in \mathbb{Z} \setminus \{0\}$. Note that if all $a^3, b^3, c^3$ have the same sign, then this is equivalent to FLT. If they don't all have the same sign, then one of them has the opposite sign as the others. Then simply re-arrange so that the one with the different sign is isolated on one side of the equation, and by renaming variables we will have arrived at the exact same statement as FLT with the natural numbers.