A prime number p such that 2p + 1 is also prime; the number 2p + 1 is called the corresponding safe prime.

Principle

The organizing idea is a paired primality relation: a base prime p is recognized by the primality of a linear function 2p + 1, creating linked prime pairs useful in multiplicative and residue arguments.

Demonstration

Concrete example: p = 11 is prime and 2·11 + 1 = 23 is prime, so 11 is a Sophie Germain prime and 23 the corresponding safe prime.

Misapplication

Credible misuse is to assume that if 2p + 1 is prime then p must be Sophie Germain in any context without checking p itself is prime; applying the concept to composite bases (e.g., treating 9 because 2·9+1=19 is prime) is invalid.

Consequence

Correct identification gives paired primes that appear in constructions of safe cyclic groups, constrains possible orders in multiplicative groups modulo safe primes, and informs combinatorial and cryptographic parameter choices.

Reversal

The inverse concept is a prime q for which (q−1)/2 is prime (a safe prime), emphasizing the larger element of the pair rather than the smaller; not every safe prime's half is necessarily prime unless checked.

Boundary

Scope: p must be an integer prime and 2p + 1 must be prime. Excludes composite p and definitions over nonintegral domains; does not assert anything about density or infinitude of such primes.

Semantic Tension

Tension arises with the term 'safe prime' (the larger partner) and with other special prime types (twin, Sophie Germain, safe) — one must distinguish which element of the pair is being named and which property (p prime versus 2p+1 prime) is primary.

Synthesis

A Sophie Germain prime is the smaller member p of a linked prime pair (p, 2p+1) where both are prime; the notion captures a linear pairing constraint that is algebraically and arithmetically significant while remaining narrow in scope and conditional on primality of both terms.