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Since same decades it is known that there are no positive integer solutions for $a^n+b^n=c^n$ for $n \gt 2$.

What is known if we see also complex integers (complex numbers with an integer real and imaginary part)?

peterh
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    The question Will Fermat's last theorem hold true on the Complex Plane? isn't a duplicate of this one, but the answer there addresses this question, namely that the problem is open. See also this MO thread. – Noah Schweber Nov 10 '23 at 20:32
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    Following links there is a mathoverflow result. What I understand: 1) it is active, hot topic today 2) there are partial results and breakthroughs 3) mathematicians believe the answer is that "FLT holds for Q(i)", meaning that Wiles' Theorem holds for gaussian integers, but we are probably from some years away from a proof. – peterh Nov 10 '23 at 21:32
  • Yes, that's my (highly non-expert) understanding as well. – Noah Schweber Nov 10 '23 at 21:38
  • peterh - shouldn't that be "FLT holds for Z(i)" rather than Q(i)? – John Nov 11 '23 at 04:00
  • @John They say Q. I think, Z(i) is what for us interesting. But I afaik these two are eqvuivalent, only the Q non-intuitive for you. I think, they might use Q because the whole Wiles-proof is based on some newly found parallelism with the elliptic curves. Do not ask the details, I am very layman :-) But the important thing is that play with curves and not integers in this proof. – peterh Nov 11 '23 at 04:12
  • @John Again something what is so trivial for a mathematician that they have even forgotten to mention it. We need to think a little about. But, afaik, FLT for Q(i) is equivalent with FLT for Z(i). You can trivially see that in both direction. But, what is important, that Q(i) plays better with curves because it likes continuous things while Z(i) does not, and Wiles' solution is using some, for us esoterically complicated correlation with elliptic curves. – peterh Nov 13 '23 at 19:15

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