Sophie Germain Prime

A Sophie Germain prime is a prime number p such that two times p plus one is also prime, so that, for example, two, three, five, eleven, twenty-three and twenty-nine are all Sophie Germain primes because doubling and adding one to each again produces a prime. The class is named after the French mathematician Sophie Germain, who in correspondence and a partial proof from the 1820s used primes with this property to make progress on a special case of Fermat's Last Theorem, showing that the theorem holds for such primes under certain conditions well before the full theorem was proved. The related prime produced by the doubling operation is called a safe prime, a term that later became important in cryptography, since safe primes are used to construct groups in which certain hard computational problems, including the discrete logarithm problem, resist known attacks. It remains an open question, part of a broader family of unresolved prime-counting conjectures, whether infinitely many Sophie Germain primes exist.