Sophie Germain and Safe Primes

Sophie Germain prime is a special type of prime number that has a special property: if 'p' is a prime, and '2p + 1' is also prime, then 'p' is called a Sophie Germain prime. The number 2p + 1 is referred to as a safe prime.

For Example: For p = 11, then 2 × 11 + 1 = 23, and since both 11 and 23 are prime, 11 is a Sophie Germain prime.

In simple words, a number 'p' is said to be a Sophie Germain Prime Number. If both p and '2p + 1' are prime numbers.

Examples of Sophie Germain Primes

Some examples of Sophie Germain primes:

p = 2, because 2 × 2 + 1 = 5 (both 2 and 5 are primes)
p = 3, because 2 × 3 + 1 = 7 (both 3 and 7 are primes)
p = 5, because 2 × 5 + 1 = 11 (both 5 and 11 are primes)
p = 11, because 2 × 11 + 1 = 23 (both 11 and 23 are primes)

Largest Sophie Germain Prime Number

As of 11 October 2024, the largest known Sophie Germain Prime Number is 2,618,163,402,417 × 2^1,290,000 − 1, which was discovered in 2016 and has digits.

Rank	Sophie Germain Prime	No of Digits	Discovered In
1	2,618,163,402,417 × 2^1,290,000 − 1	388,342	2016
2	18,543,637,900,515 × 2^666,667 − 1	200,701	2012

Safe Prime Numbers

A safe prime is a type of prime number that can be expressed in the form of 2p + 1 where p is also a prime number. Here 'p' is called a Sophie Germain Prime Number.

For Example: 47 can be expressed as, 47 = 2 × 23 + 1, where 23 is a prime number. Thus, 47 is a Safe prime number.

In other words we can also say that a number 'x' is said to be a Safe Prime Number. if both x and \frac{x - 1}{2} are prime numbers.

Examples of Safe Primes

Some examples of Safe primes:

p = 11, because 2 × 5 + 1 = 11 (both 11 and 5 are primes).
p = 23, because 2 × 11 + 1 = 23 (both 23 and 11 are primes).
p = 59, because 2 × 23 + 1 = 59 (both 59 and 23 are primes).

Largest Safe Prime Number

As of 11 October 2024 the largest known Safe Prime Number is 2,618,163,402,417 × 2^1,290,001 − 1.

List of first 20 Sophie Germain and Safe Primes

S No.	Sophie Germain Prime	Safe Prime	S No.	Sophie Germain Prime	Safe Prime
1	2	5	11	113	227
2	3	7	12	131	263
3	5	11	13	173	347
4	11	23	14	179	359
5	23	47	15	191	383
6	29	59	16	233	467
7	41	83	17	239	479
8	53	107	18	251	503
9	83	167	19	281	563
10	89	179	20	293	587

Circular Prime
Absolute Prime
Deletable Prime
Cullen Primes
Fermat Primes
Pallindormic Primes

S No.	Sophie Germain Prime	Safe Prime	S No.	Sophie Germain Prime	Safe Prime
1	2	5	11	113	227
2	3	7	12	131	263
3	5	11	13	173	347
4	11	23	14	179	359
5	23	47	15	191	383
6	29	59	16	233	467
7	41	83	17	239	479
8	53	107	18	251	503
9	83	167	19	281	563
10	89	179	20	293	587

S No.	Sophie Germain Prime	Safe Prime	S No.	Sophie Germain Prime	Safe Prime
1	2	5	11	113	227
2	3	7	12	131	263
3	5	11	13	173	347
4	11	23	14	179	359
5	23	47	15	191	383
6	29	59	16	233	467
7	41	83	17	239	479
8	53	107	18	251	503
9	83	167	19	281	563
10	89	179	20	293	587