KnowraSophie Germain primeSophie Germain primeA prime number p for which 2p + 1 is also prime. The associated prime 2p + 1 is called a safe prime.BriefConnectPrime number: A natural number greater than one whose only positive divisors are one and itself. The defining condition requires p itself to be prime.Fermat number: An integer of the form 2ⁿ + 1, where n is a nonnegative integer. For odd Sophie Germain primes p, the number 2p + 1 divides 2ᵖ + 1.Sophie Germain: A French mathematician whose work included number theory, elasticity, and Fermat's Last Theorem. The primes take her name because of her work on Fermat's Last Theorem.Diffie–Hellman key exchange: A protocol that lets two parties derive a shared secret over a public channel using modular arithmetic. Safe primes derived from Sophie Germain primes are often used to choose groups for key exchange.Safe prime: A prime number q such that (q − 1)/2 is also prime. Every Sophie Germain prime produces a safe prime by q = 2p + 1.Sophie Germain identity: The factorization a⁴ + 4b⁴ = (a² − 2ab + 2b²)(a² + 2ab + 2b²). Its factorization gives a classic obstruction to primes of the form x⁴ + 4y⁴.Fermat's Last Theorem: The theorem that xⁿ + yⁿ = zⁿ has no positive-integer solutions for n greater than two. Germain developed a significant approach to special exponents using properties of primes.Finite cyclic group: A finite group generated by repeated applications of one element. The multiplicative groups modulo safe primes have simple orders useful in cryptographic constructions.Twin prime: A pair of prime numbers that differ by two. Both concern linked prime pairs, but their defining gaps and equations differ.Quadratic reciprocity: A theorem describing when one odd prime is a square modulo another. Residue constraints from quadratic reciprocity can restrict prime patterns involving p and 2p + 1.Show all 20Linked from 5 pagesSophie Germain's theoremBroader topic: These are exactly the exponents covered by the theorem.Show all 5