Sophie Germain primes

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Sophie Germain primes are prime numbers $p$ such that $2p+1$ is also prime. For example, 23 is a Sophie Germain prime, since twice 23 is 46, one less than 47, also prime. See A005384. Compare safe primes. It is not known if there are infinitely many Sophie Germain primes.

These primes are named after the mathematician Sophie Germain, who, in her attempt to prove Fermat's last theorem, proved it for many specific cases.^[1]

Theorem. If $p$ is an odd prime such that $2p+1$ is also prime, then the equation $x^{p}+y^{p}=z^{p}$ has no solution in integers.^[2]

Proof. PROOF GOES HERE AND ALSO IN THE CORRESPONDING SPOT IN Fermat's last theorem. END OF PROOF MARK GOES HERE

↑ Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): p. 5.
↑ Note that although technically 2 is a Sophie Germain prime, for Germain's purpose it did not count, since $x^{2}+y^{2}=z^{2}$ has infinitely many solutions.

[1] Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): p. 5.

[2] Note that although technically 2 is a Sophie Germain prime, for Germain's purpose it did not count, since $x^{2}+y^{2}=z^{2}$ has infinitely many solutions.

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Sophie Germain primes

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