Definition:Sophie Germain Prime

Definition

A Sophie Germain prime is a prime number $p$ such that $2 p + 1$ is also prime.

It follows from this definition that $2 p + 1$ is a safe prime.

Sequence

The sequence of Sophie Germain primes begins:

$2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, \ldots$

Largest Known

The largest known Sophie Germain primes (as of May 2017) are as follows:

Sophie Germain prime	Number of digits	Date discovered	Discovered by
$137211941292195 \times 2^{171960} - 1$	$51780$	3rd May 2006	Zoltán Járai, Gabor Farkas, Timea Csajbok, János Kasza and Antal Járai
$48047305725 \times 2^{172403} - 1$	$51910$	25th January 2007	David Underbakke
$607095 \times 2^{176311} - 1$	$53081$	18th September 2009	Tom Wu
$99064503957 \times 2^{200008} - 1$	$60220$	April 2016	S. Urushihata
$620366307356565 \times 2^{253824} - 1$	$76424$	2nd November 2009	Zoltán Járai, Gabor Farkas, Timea Csajbok, János Kasza and Antal Járai
$648621027630345 \times 2^{253824} - 1$	$76424$	2nd November 2009	Zoltán Járai, Gabor Farkas, Timea Csajbok, János Kasza and Antal Járai
$183027 \times 2^{265440} - 1$	$79911$	22nd March 2010	Tom Wu
$18543637900515 \times 2^{666667} - 1$	$200701$	April 2012	Philipp Bliedung
$2618163402417 \times 2^{1290000} - 1$	$388342$	February 2016	James Scott Brown

Also see

• Results about Sophie Germain primes can be found here.

Source of Name

This entry was named for Marie-Sophie Germain.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $89$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $89$

• Weisstein, Eric W. "Sophie Germain Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SophieGermainPrime.html