Sophie Germain primes

A prime number $p$

is a Sophie Germain prime if $2p+1$

is also a prime.

The name originates from the fact that in 1823 Sophie Germain proved that one subcase of the Fermat Last Theorem holds for all the prime exponents $p$ such that $2p+1$ is also prime.

It is conjectured that there are infinitely many Sophie Germain primes, and that up to $n$ there are approximately

$2C\frac{n}{(\ln n)^2} \approx 1.32\frac{n}{(\ln n)^2}$

such primes, where

$C =\prod_{p>2}\frac{p(p-2)}{(p-1)^2}\approx 0.66016$

(in which the product is over all odd primes $p$

) is the so-called Hardy–Littlewood's twin prime constant.

Up to $10^{12}$ there are 1822848478 Sophie Germain primes, and the formula above underestimates the actual number by about 5.4%.

The first Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

Useful links Wikipedia, Sophie Germain prime
Mathworld, Sophie Germain Prime
OEIS, Sequence A005384

Sophie Germain primes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 293 1409 2039 5333 + 9999555551 aban 11 23 29 41 53 + 10000000343 alternating 23 29 41 83 89 + 989898341 amenable 29 41 53 89 113 + 999998801 apocalyptic 251 443 1019 1031 1103 + 29873 arithmetic 11 23 29 41 53 + 9999653 balanced p. 53 173 593 653 1103 + 9999994883 bemirp 1601 1901 10061 10091 16001 + 1988809001 c.decagonal 11 281 911 1901 15401 + 9892129601 c.heptagonal 953 5741 10781 14561 46691 + 9956364461 c.square 41 113 761 1013 14621 + 9997687013 canyon 419 509 659 719 809 + 9876543179 Chen 11 23 29 41 53 + 99999611 congruent 23 29 41 53 173 + 9999653 Cunningham 1601 12101 25601 28901 62501 + 9960040001 Curzon 29 41 53 89 113 + 199999853 cyclic 11 23 29 41 53 + 9999653 d-powerful 89 2063 2753 2939 4373 + 9978833 de Polignac 251 509 809 1019 1973 + 99998999 deficient 11 23 29 41 53 + 9999653 dig.balanced 11 41 653 659 809 + 199993361 economical 11 23 29 41 53 + 19999379 emirp 113 179 359 743 761 + 199999661 equidigital 11 23 29 41 53 + 19999379 esthetic 23 89 12101 212123 234323 + 6787654343 Eulerian 11 1013 evil 23 29 53 83 89 + 999998801 fibodiv 911 1499 8612801 916666661 Fibonacci 89 233 Friedman 12101 15629 28559 29531 32771 + 953321 Gilda 29 683 good prime 11 29 41 53 179 + 199883693 happy 23 239 293 653 683 + 9999299 Honaker 131 1049 2141 2273 5711 + 999821261 hungry 2003 iban 11 23 41 173 443 + 777173 inconsummate 173 431 443 491 761 + 999623 Jacobsthal 11 683 43691 2796203 junction 113 509 719 911 1013 + 99996749 katadrome 41 53 83 431 641 + 986543 Kynea 23 Leyland 593 lonely 23 53 24281 38501 10938023 Lucas 11 29 3010349 54018521 m-pointer 23 1111361 3221111 9111341 121235111 + 4111311131 magnanimous 11 23 29 41 83 + 48804809 metadrome 23 29 89 179 239 + 145679 modest 23 29 89 233 509 + 1997009009 mountain 131 173 191 251 281 + 5698765421 nialpdrome 11 41 53 83 431 + 9999988811 nude 11 oban 11 23 29 53 83 + 953 odious 11 41 131 173 179 + 999999191 Ormiston 1931 34631 37379 63179 66431 + 1999995713 palindromic 11 131 191 12821 14741 + 999212999 palprime 11 131 191 12821 14741 + 999212999 pancake 11 29 191 5051 6329 + 9991960931 panconsummate 11 23 53 89 239 pandigital 11 partition 11 pernicious 11 41 131 173 179 + 9999299 Perrin 29 43721 plaindrome 11 23 29 89 113 + 7777778999 prime 11 23 29 41 53 + 10000000343 primeval 113 1013 1003679 100234679 Proth 41 113 641 1409 1601 + 9996206081 repdigit 11 repunit 28792661 78914411 943280801 7294932341 self 53 233 659 1223 1289 + 999998903 self-describing 10311533 12313319 15333119 16331831 17143331 + 4441411913 sliding 11 29 641 strobogrammatic 11 16091 1881881 6801089 16969691 + 6988968869 strong prime 11 29 41 179 191 + 99999611 super-d 131 281 419 431 719 + 9998903 tetranacci 29 trimorphic 251 74218751 truncatable prime 23 29 53 83 113 + 9439663853 twin 11 29 41 179 191 + 999999191 uban 11 23 29 41 53 + 9096000029 Ulam 11 53 131 431 1103 + 9991259 undulating 131 191 upside-down 1289 1559 7193 7643 7823 + 9993557111 weak prime 23 83 89 113 131 + 99999563 weakly prime 3326489 5575259 7469789 9537371 14075273 + 9999130709 Woodall 23 191 4373 590489 Zuckerman 11 zygodrome 11 44111 77711 88811 1117799 + 9999988811