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

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In summary, Fermat's last theorem (FLT) states that there are no non-trivial integer solutions to the equation x^n+y^n=z^n for any integer n greater than 2. This can be reduced to proving the case when n is prime or 4. The case n=4 is already proven, but the proof for primes of the form 3k+1 is different than for primes of the form 3k+2.
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smslca
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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*
 

1. What is Fermat's last theorem?

Fermat's last theorem is a mathematical conjecture proposed by Pierre de Fermat in 1637. It states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.

2. Has Fermat's last theorem been proven?

Yes, Fermat's last theorem has been proven by Andrew Wiles in 1994 after over 350 years of attempts by mathematicians. Wiles used advanced mathematical techniques, including modular forms and elliptic curves, to prove the theorem.

3. Why is Fermat's last theorem significant?

Fermat's last theorem is significant because it is one of the most famous and long-standing problems in mathematics. It has also inspired the development of new mathematical fields and techniques, such as algebraic number theory and elliptic curves.

4. Who was Pierre de Fermat?

Pierre de Fermat was a French mathematician, lawyer, and government official in the 17th century. He is best known for his contributions to number theory, including Fermat's last theorem, and his work in analytic geometry.

5. Are there any variations or generalizations of Fermat's last theorem?

Yes, there are several variations and generalizations of Fermat's last theorem, such as the Beal conjecture, which asks for solutions to an + bn = cn where a, b, and c are positive integers and n is a real number greater than 2. There are also generalizations to other number systems, such as complex numbers and Gaussian integers.

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