Editing Safe and Sophie Germain primes (section)

==Individual numbers==
The first few Sophie Germain primes (those less than 1000) are

:2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ... {{OEIS2C|id=A005384}}

Hence, the first few safe primes are

:5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, ... {{OEIS2C|id=A005385}}

In [[cryptography]], much larger Sophie Germain primes like 1,846,389,521,368 + 11<sup>600</sup> are required.

Two distributed computing projects, [[PrimeGrid]] and [[Twin Prime Search]], include searches for large Sophie Germain primes. Some of the largest known Sophie Germain primes are given in the following table.<ref>[http://primes.utm.edu/top20/page.php?id=2 The Top Twenty Sophie Germain Primes]&nbsp;— from the [[Prime Pages]]. Retrieved 17 May 2020.</ref>
{| class="wikitable"
|-
! Value !! Number<br>of&nbsp;digits !! Time of<br>discovery !! Discoverer
|-
| 2618163402417 × 2<sup>1290000</sup> − 1 || align="right" | 388342 || February 2016 || Dr. James Scott Brown in a distributed [[PrimeGrid]] search using the programs TwinGen and [[Lucas–Lehmer–Riesel test|LLR]]<ref>{{cite web|title=PrimeGrid's Sophie Germain Prime Search|url=https://www.primegrid.com/download/SGS_2618163402417_1290000.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.primegrid.com/download/SGS_2618163402417_1290000.pdf |archive-date=2022-10-09 |url-status=live|publisher=PrimeGrid|access-date=29 February 2016}}</ref>
|-
| 18543637900515 × 2<sup>666667</sup> − 1 || align="right" | 200701 || April 2012 || Philipp Bliedung in a distributed [[PrimeGrid]] search using the programs TwinGen and [[Lucas–Lehmer–Riesel test|LLR]]<ref>{{cite web|title=PrimeGrid's Sophie Germain Prime Search|url=http://www.primegrid.com/download/SGS_666667.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.primegrid.com/download/SGS_666667.pdf |archive-date=2022-10-09 |url-status=live|publisher=PrimeGrid|access-date=18 April 2012}}</ref>
|-
|183027 × 2<sup>265440</sup> − 1 || align="right" | 79911 || March 2010 || Tom Wu using LLR<ref>[http://primes.utm.edu/primes/page.php?id=92222 The Prime Database: 183027*2^265440-1]. From The [[Prime Pages]].</ref>
|-
|style="white-space:nowrap"|648621027630345 × 2<sup>253824</sup> − 1 and<br>620366307356565 × 2<sup>253824</sup> − 1 || align="right" | 76424 ||style="white-space:nowrap"| November 2009 || Zoltán Járai, Gábor Farkas, Tímea Csajbók, János Kasza and Antal Járai<ref>[http://primes.utm.edu/primes/page.php?id=90907 The Prime Database: 648621027630345*2^253824-1].</ref><ref>[http://primes.utm.edu/primes/page.php?id=90711 The Prime Database: 620366307356565*2^253824-1]</ref>
|-
|1068669447 × 2<sup>211088</sup> − 1 || align="right" | 63553 || May 2020 || Michael Kwok<ref>[https://primes.utm.edu/primes/page.php?id=130903 The Prime Database: 1068669447*2^211088-1] From The [[Prime Pages]].</ref>
|-
|99064503957 × 2<sup>200008</sup> − 1 || align="right" | 60220 || April 2016 || S. Urushihata<ref>[https://primes.utm.edu/primes/page.php?id=121507 The Prime Database: 99064503957*2^200008-1] From The [[Prime Pages]].</ref>
|-
|607095 × 2<sup>176311</sup> − 1 || align="right" | 53081 ||style="white-space:nowrap"| September 2009 || Tom Wu<ref>[http://primes.utm.edu/primes/page.php?id=89999 The Prime Database: 607095*2^176311-1].</ref>
|-
|48047305725 × 2<sup>172403</sup> − 1 || align="right" | 51910 || January 2007 || David Underbakke using TwinGen and LLR<ref>[http://primes.utm.edu/primes/page.php?id=79261 The Prime Database: 48047305725*2^172403-1].</ref>
|-
|style="white-space:nowrap"|137211941292195 × 2<sup>171960</sup> − 1 || align="right" | 51780 || May 2006 || Járai et al.<ref>[http://primes.utm.edu/primes/page.php?id=77705 The Prime Database: 137211941292195*2^171960-1].</ref>
|-
|}

On 2 Dec 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé, and Paul Zimmermann announced the computation of a [[discrete logarithm]] modulo the 240-digit (795 bit) prime RSA-240 + 49204 (the first safe prime above RSA-240) using a [[General number field sieve|number field sieve]] algorithm; see [[Discrete logarithm records]].