Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than two.
Now the question is will Fermat's last theorem hold true if we extend the question to the complex plane. Ie when $a$, $b$ or $c$ can be complex numbers. Why or Why not and is there any prove to it?
Note that for any $w\in\mathbb{C}$, with $w\not=0$, the equation $z^n=w$ has $n$ distinct complex roots. So Fermat's Last Theorem does not hold in $\mathbb{C}$. For example $$(3+i\sqrt{7})^4 + 4^4=(1-i\sqrt{7})^4.$$ Moreover, even restricting to real numbers, we have that $$1^n+1^n=2=(\sqrt[n]{2})^n$$ On the other hand, Fermat's Last Theorem for Gaussian Integers is still open: see this question.