## Sophie Germain (p) |

If both *p* and 2*p*+1 are prime, then
*p* is a Sophie Germain prime.
The first few Sophie Germain primes are 2, 3, 5, 11,
23, 29, 41, 53, 83, 89, 113, and 131. Around 1825 Sophie
Germain proved that the first case of Fermat's Last
Theorem is true for such primes. Soon after Legendre
began to generalize this by showing the first case of
FLT also holds for odd primes *p* such that
*kp*+1 is prime, *k*=4, 8, 10, 14 and 16.
In 1991 Fee and Granville [FG91]
extended this to *k*__<__100, *k* not
a multiple of three. Many similar results were
also shown, but now that Fermat's Last Theorem
has been proven by Wiles, they are of less
interest.
_{2} is the twin
prime constant (estimated by Wrench and others to
be approximately 0.6601618158...).
This estimate works suprisingly well! For example:

Are there infinitely many Sophie Germain primes?
Ribenboim indicates that the sieve methods of Brun
(see the twin primes page) can be
used to estimate that the number of primes *p* <
*x* for
which *kp*+*a* is prime is bounded above by
C x/(log *x*)^{2}
(so they have density zero among the primes).
Heuristically, it seems reasonable to conjecture
that there is a lower bound of this form as well.
More specifically (see
a simple
heuristic), it is conjectured that the number of
Sophie Germain primes less than
*N* is asympototic to

where C

N | actual | estimate |
---|---|---|

1,000 | 37 | 39 |

100,000 | 1171 | 1166 |

10,000,000 | 56032 | 56128 |

100,000,000 | 423140 | 423295 |

1,000,000,000 | 3308859 | 3307888 |

10,000,000,000 | 26569515 | 26568824 |

Euler and
Lagrange
proved that if we also have *p* ≡ 3
(mod 4) and *p* > 3, then
2*p*+1 is prime (and *p* is a Sophie Germain prime)
if and only if 2*p*+1
divides the Mersenne M_{p}.

(Thanks to Chip Kerchner for the last two entries in the table above.)

>rank prime digits who when comment 1 2618163402417 · 2^{1290000}- 1388342 L927 Feb 2016 Sophie Germain (p) 2 18543637900515 · 2^{666667}- 1200701 L2429 Apr 2012 Sophie Germain (p) 3 183027 · 2^{265440}- 179911 L983 Mar 2010 Sophie Germain (p) 4 648621027630345 · 2^{253824}- 176424 x24 Nov 2009 Sophie Germain (p) 5 620366307356565 · 2^{253824}- 176424 x24 Nov 2009 Sophie Germain (p) 6 1068669447 · 2^{211088}- 163553 L4166 May 2020 Sophie Germain (p) 7 99064503957 · 2^{200008}- 160220 L95 Apr 2016 Sophie Germain (p) 8 607095 · 2^{176311}- 153081 L983 Sep 2009 Sophie Germain (p) 9 48047305725 · 2^{172403}- 151910 L99 Jan 2007 Sophie Germain (p) 10 137211941292195 · 2^{171960}- 151780 x24 May 2006 Sophie Germain (p) 11 21195711 · 2^{143630}- 143245 L3494 Jun 2019 Sophie Germain (p) 12 838269645 · 2^{143165}- 143106 L3494 Jun 2019 Sophie Germain (p) 13 570409245 · 2^{143163}- 143106 L3494 Jun 2019 Sophie Germain (p) 14 2830598517 · 2^{143112}- 143091 L3494 Jul 2019 Sophie Germain (p) 15 4158932595 · 2^{143073}- 143079 L3494 Jul 2019 Sophie Germain (p) 16 31737014565 · 2^{140003}- 142156 L95 Dec 2010 Sophie Germain (p) 17 14962863771 · 2^{140001}- 142155 L95 Dec 2010 Sophie Germain (p) 18 13375563435 · 2^{137136}- 141293 p364 Jan 2018 Sophie Germain (p) 19 10429091973 · 2^{135135}- 140690 p364 Jan 2018 Sophie Germain (p) 20 73378515705 · 2^{133147}- 140093 L167 Jan 2018 Sophie Germain (p)

- The World of mathematics': Sophie Germain Primes
- The Prime Glossary': Sophie Germain prime
- The chronology of prime number records' Sophie prime records by year

- Agoh2000
Agoh, Takashi, "On Sophie Germain primes,"Tatra Mt. Math. Publ.,20(2000) 65--73. Number theory (Liptovský Ján, 1999).MR 1845446- CFJJK2006
Csajbók, T.,Farkas, G.,Járai, A.,Járai, Z.andKasza, J., "Report on the largest known Sophie Germain and twin primes,"Ann. Univ. Sci. Budapest. Sect. Comput.,26(2006) 181--183.MR 2388687- Dubner96
H. Dubner, "Large Sophie Germain primes,"Math. Comp.,65:213 (1996) 393--396.MR 96d:11008(Abstract available)- JR2007
Jaroma, John H.andReddy, Kamaliya N., "Classical and alternative approaches to the Mersenne and Fermat numbers,"Amer. Math. Monthly,114:8 (2007) 677--687.MR 2354438- Peretti1987
Peretti, A., "The quantity of Sophie Germain primes less thanx,"Bull. Number Theory Related Topics,11:1-3 (1987) 81--92.MR 995537- Ribenboim95
P. Ribenboim,The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5.MR 96k:11112[An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]- Yates1987
Yates, Samuel,Sophie Germain primes. In "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991. pp. 882--886,MR 1146271

Chris K. Caldwell
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