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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?

Bernard
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ys wong
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1 Answers1

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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.

Arturo Magidin
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Robert Z
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  • @ArturoMagidin My apologies. I will make it a comment. – Tito Piezas III Jan 04 '24 at 04:27
  • Note that for $n=4$, $$ 4^4 + (-3-\sqrt{-7})^4 = (1-\sqrt{-7})^4 $$ and for $n=5$, $$ 2^5 + (-1-\sqrt{-3})^5 = (1-\sqrt{-3})^5 $$ then these solutions to $x^n+y^n = z^n$ also have $x+y = z$ and considered trivial by the "Debarre-Klassen conjecture", discussed in the linked 2012 MO question above. – Tito Piezas III Jan 04 '24 at 04:30