# Sophie Germain (p)

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

This page is about one of those forms.

### Definitions and Notes

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.

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

_{2}is the twin prime constant (estimated by Wrench and others to be approximately 0.6601618158...). This estimate works suprisingly well! For example:

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.)

### Record Primes of this Type

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 22942396995 · 2^{265776}- 180017 L3494 Nov 2023 Sophie Germain (p) 4 183027 · 2^{265440}- 179911 L983 Mar 2010 Sophie Germain (p) 5 648621027630345 · 2^{253824}- 176424 x24 Nov 2009 Sophie Germain (p) 6 620366307356565 · 2^{253824}- 176424 x24 Nov 2009 Sophie Germain (p) 7 10957126745325 · 2^{222333}- 166942 L5843 Nov 2023 Sophie Germain (p) 8 20690306380455 · 2^{222332}- 166942 L5843 Nov 2023 Sophie Germain (p) 9 10030004436315 · 2^{222333}- 166942 L5843 Nov 2023 Sophie Germain (p) 10 8964472847055 · 2^{222333}- 166942 L5843 Nov 2023 Sophie Germain (p) 11 1068669447 · 2^{211088}- 163553 L4166 May 2020 Sophie Germain (p) 12 99064503957 · 2^{200008}- 160220 L95 Apr 2016 Sophie Germain (p) 13 12443794755 · 2^{184516}- 155555 L3494 Sep 2021 Sophie Germain (p) 14 21749869755 · 2^{184515}- 155555 L3494 Sep 2021 Sophie Germain (p) 15 14901867165 · 2^{184515}- 155555 L3494 Sep 2021 Sophie Germain (p) 16 607095 · 2^{176311}- 153081 L983 Sep 2009 Sophie Germain (p) 17 48047305725 · 2^{172403}- 151910 L99 Jan 2007 Sophie Germain (p) 18 137211941292195 · 2^{171960}- 151780 x24 May 2006 Sophie Germain (p) 19 4931286045 · 2^{152849}- 146022 L5389 Jul 2021 Sophie Germain (p) 20 4318624617 · 2^{152849}- 146022 L5389 Jul 2021 Sophie Germain (p)

### Related Pages

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

### References

- 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