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In number theory, a prime number ''p'' is a Sophie Germain prime if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a safe prime. For example, 29 is a Sophie Germain prime and 2 × 29 + 1 = 59 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem.〔Specifically, Germain proved that the first case of Fermat's Last Theorem, in which the exponent divides one of the bases, is true for every Sophie Germain prime, and she used similar arguments to prove the same for all other primes up to 100. For details see .〕 Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven. ==Individual numbers== The first few Sophie Germain primes are: (less than 1000) :2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ... . In cryptography much larger Sophie Germain primes like are required. Two distributed computing projects, PrimeGrid and Twin Prime Search, include searches for large Sophie Germain primes. The largest known Sophie Germain primes are:〔(The Top Twenty Sophie Germain Primes ) — from the Prime Pages. Retrieved 24 April 2015.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sophie Germain prime」の詳細全文を読む スポンサード リンク
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