Can Fermat's Last Theorem be proven for primes of the form 3k+1 and 3k+2?

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Fermat's Last Theorem asserts that there are no non-trivial integer solutions to the equation x^n + y^n = z^n for integers n greater than 2. The discussion highlights the relevance of prime numbers of the forms 3k+1 and 3k+2 in proving the theorem, noting that the proof for these primes differs significantly. It is suggested that proving the theorem for these specific primes could simplify the overall proof. The case for n=4 is mentioned as a classical example that is already established. The conversation emphasizes the complexity of the proof related to different prime forms.
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fermat's last theorem?

What is Fermat's last theorem?
and How is that related to primes numbers of form 3x+1 and 3x+2
 
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FLT states: If n is an integer greater than 2, then there are no non-trivial integer solutions to the equation x^n+y^n=z^n.

One reduction for the proof is that it would be enough to prove the case when n is prime (or 4). The case n=4 is classical. It turns out that argument for primes of the form 3k+1 is different than for primes of the form 3k+2.

*warning, Latex seems out of whack today*
 

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