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Los números primos: parejas twin primes.
Sophie Germain primes are named after French mathematician Sophie Germain , who used them in her investigations of Fermat's Last Theorem.
It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven. It is conjectured that there are infinitely many Sophie Germain primes, but this has not been proven. A heuristic estimate for the number of Sophie Germain primes less than n is . The form of this estimate is due to G. Littlewood , who applied a similar estimate to twin primes. Every term of such a sequence except the last is a Sophie Germain prime, and every term except the first is a safe prime.
Extending the conjecture that there exist infinitely many Sophie Germain primes, it has also been conjectured that arbitrarily long Cunningham chains exist,  although infinite chains are known to be impossible. Historically, this result of Leonhard Euler was the first known criterion for a Mersenne number with a prime index to be composite.
Safe and strong primes are useful as the factors of secret keys in the RSA cryptosystem , because they prevent the system being broken by certain factorization algorithms such as Pollard's rho algorithm that would apply to secret keys formed from non-strong primes.
Similar issues apply in other cryptosystems as well, including Diffie—Hellman key exchange and similar systems that depend on the security of the discrete log problem rather than on integer factorization. In the first version of the AKS primality test paper, a conjecture about Sophie Germain primes is used to lower the worst case complexity from O log 12 n to O log 6 n.
A later version of the paper is shown to have time complexity O log 7. Sophie Germain primes may be used in the generation of pseudo-random numbers. Note that these digits are not appropriate for cryptographic purposes, as the value of each can be derived from its predecessor in the digit-stream. Sophie Germain primes are mentioned in the stage play Proof  and the subsequent film. From Wikipedia, the free encyclopedia. Parts of this article those related to section, specifically the table need to be updated.
Please update this article to reflect recent events or newly available information. Are there infinitely many Sophie Germain primes? For details see Edwards, Harold M.
Retrieved 24 April Retrieved 18 April From The Prime Pages. November 22, , Are 'strong' primes needed for RSA? When Hal Harold remembers what a Germain prime is, he speaks to Catherine in a way that would be patronizing to another mathematician.
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