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Sophie Germain (p) |
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 C2 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 2p+1 is prime (and p is a Sophie Germain prime) if and only if 2p+1 divides the Mersenne Mp.
(Thanks to Chip Kerchner for the last two entries in the table above.)
>rank prime digits who when comment 1 2618163402417 · 21290000 - 1 388342 L927 Feb 2016 Sophie Germain (p) 2 18543637900515 · 2666667 - 1 200701 L2429 Apr 2012 Sophie Germain (p) 3 183027 · 2265440 - 1 79911 L983 Mar 2010 Sophie Germain (p) 4 648621027630345 · 2253824 - 1 76424 x24 Nov 2009 Sophie Germain (p) 5 620366307356565 · 2253824 - 1 76424 x24 Nov 2009 Sophie Germain (p) 6 1068669447 · 2211088 - 1 63553 L4166 May 2020 Sophie Germain (p) 7 99064503957 · 2200008 - 1 60220 L95 Apr 2016 Sophie Germain (p) 8 607095 · 2176311 - 1 53081 L983 Sep 2009 Sophie Germain (p) 9 48047305725 · 2172403 - 1 51910 L99 Jan 2007 Sophie Germain (p) 10 137211941292195 · 2171960 - 1 51780 x24 May 2006 Sophie Germain (p) 11 21195711 · 2143630 - 1 43245 L3494 Jun 2019 Sophie Germain (p) 12 838269645 · 2143165 - 1 43106 L3494 Jun 2019 Sophie Germain (p) 13 570409245 · 2143163 - 1 43106 L3494 Jun 2019 Sophie Germain (p) 14 2830598517 · 2143112 - 1 43091 L3494 Jul 2019 Sophie Germain (p) 15 4158932595 · 2143073 - 1 43079 L3494 Jul 2019 Sophie Germain (p) 16 31737014565 · 2140003 - 1 42156 L95 Dec 2010 Sophie Germain (p) 17 14962863771 · 2140001 - 1 42155 L95 Dec 2010 Sophie Germain (p) 18 13375563435 · 2137136 - 1 41293 p364 Jan 2018 Sophie Germain (p) 19 10429091973 · 2135135 - 1 40690 p364 Jan 2018 Sophie Germain (p) 20 73378515705 · 2133147 - 1 40093 L167 Jan 2018 Sophie Germain (p)
- 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. and Kasza, 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. and Reddy, Kamaliya N., "Classical and alternative approaches to the Mersenne and Fermat numbers," Amer. Math. Monthly, 114:8 (2007) 677--687. MR 2354438
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- Peretti, A., "The quantity of Sophie Germain primes less than x," Bull. Number Theory Related Topics, 11:1-3 (1987) 81--92. MR 995537
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- 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