Sophie Germain proved Fermat's Last Theorem $x^p+y^p \neq z^p$ for the special case where p is a Sophie Germain prime and $p\not|xyz$. Does any one know of a proof for the other case, where $p|xyz$? Note: I am looking for a proof restricted to the Sophie Germain primes, as of course, Wiles proved this generally.
Fermat's Last Theorem - Special Case of Sophie Germain Primes
14
$\begingroup$
number-theory
reference-request
-
1This is a natural question to ask which does not seem to have an answer. – 2011-02-18
1 Answers
12
These two cases are traditionally called ${\bf Case \,\, 1}$ and ${\bf Case \,\, 2}$, and you are after a proof of Case 2.
The essence of Dirichlet's proof of Case 2 when $p=5$ can be found in Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, by Harold M. Edwards. It's in chapter three, at about page 70.
-
0Thank you , Dr Jennings. Do you know of a proof for Case 2 for all Sophie Germain primes? – 2011-02-18
-
2@James: There's always Wiles' proof but, of course, that encompasses more than Sophie Germain primes. Sorry, I don't know of a proof restricted only to Sophie Germain primes in Case 2 but I'm fairly sure that the answer to your question is that one does not exist. – 2011-02-18
-
0Hmm, I was afraid of that. Thank you anyway, I will check out the Dirichlet proof. – 2011-02-18