Exercise 1144Number theory › PrimesOlympiadAdded 3 Nov 2026

Sophie Germain primes

#Sophie Germain#primes#examples

Statement

Define Sophie Germain primes and list all such $p$ with $p<50$ .

Hint

$p$ is Sophie Germain if both $p$ and $2p+1$ are prime.

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Definition. $p$ is Sophie Germain if $p$ and $2p+1$ are both prime.

List under 50.
- $p=2$ : $2p+1=5$ ✓
- $p=3$ : $7$ ✓
- $p=5$ : $11$ ✓
- $p=11$ : $23$ ✓
- $p=23$ : $47$ ✓
- $p=29$ : $59$ ✓
- $p=41$ : $83$ ✓

Conclusion. Seven such primes under $50$ .

Remark. Sophie Germain primes are used in cryptography; it is unknown whether infinitely many exist.