Fermat's Last Theorem - Special Case of Sophie Germain Primes

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.

• This is a natural question to ask which does not seem to have an answer. – awllower Feb 18 '11 at 14:47

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.

• Thank you , Dr Jennings. Do you know of a proof for Case 2 for all Sophie Germain primes? – James Feb 18 '11 at 10:43
• @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. – Derek Jennings Feb 18 '11 at 13:04
• Hmm, I was afraid of that. Thank you anyway, I will check out the Dirichlet proof. – James Feb 18 '11 at 14:21

protected by t.b.Apr 17 '12 at 21:15

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