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

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SUMMARY

Fermat's Last Theorem (FLT) 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 unique challenges in proving FLT for prime numbers of the forms 3k+1 and 3k+2, noting that the proof for n=4 is classical and serves as a foundation. It emphasizes that the approaches to proving FLT differ significantly between these two prime forms, necessitating distinct methodologies.

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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 [tex]x^n+y^n=z^n[/tex].

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

*warning, Latex seems out of whack today*
 

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