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 + 9999166601 9999555551 aban 11 23 29 + 9999000983 10000000343 alternating 23 29 41 + 989894981 989898341 amenable 29 41 53 + 999998789 999998801 apocalyptic 251 443 1019 + 29723 29873 arithmetic 11 23 29 + 9999299 9999653 balanced p. 53 173 593 + 9999922643 9999994883 bemirp 1601 1901 10061 + 1986199601 1988809001 c.decagonal 11 281 911 + 9830843111 9892129601 c.heptagonal 953 5741 10781 + 9929501513 9956364461 c.square 41 113 761 + 9972250313 9997687013 Chen 11 23 29 + 99999551 99999611 congruent 23 29 41 + 9998903 9999653 Cunningham 1601 12101 25601 + 9894280901 9960040001 Curzon 29 41 53 + 199999661 199999853 cyclic 11 23 29 + 9999299 9999653 d-powerful 89 2063 2753 + 9972569 9978833 de Polignac 251 509 809 + 99997619 99998999 deficient 11 23 29 + 9999299 9999653 dig.balanced 11 41 653 + 199987241 199993361 economical 11 23 29 + 19999253 19999379 emirp 113 179 359 + 199998923 199999661 equidigital 11 23 29 + 19999253 19999379 esthetic 23 89 12101 + 6765654323 6787654343 Eulerian 11 1013 evil 23 29 53 + 999998789 999998801 fibodiv 911 1499 8612801 916666661 Fibonacci 89 233 Friedman 12101 15629 28559 + 946733 953321 Gilda 29 683 good prime 11 29 41 + 199882961 199883693 happy 23 239 293 + 9998903 9999299 Honaker 131 1049 2141 + 999412313 999821261 hungry 2003 iban 11 23 41 + 777041 777173 inconsummate 173 431 443 + 998969 999623 Jacobsthal 11 683 43691 2796203 junction 113 509 719 + 99996353 99996749 katadrome 41 53 83 + 764321 986543 Kynea 23 Leyland 593 lonely 23 53 24281 38501 10938023 Lucas 11 29 3010349 54018521 m-pointer 23 1111361 3221111 + 2232113111 4111311131 magnanimous 11 23 29 + 800483 48804809 metadrome 23 29 89 + 134789 145679 modest 23 29 89 + 1994010989 1997009009 nialpdrome 11 41 53 + 9999977441 9999988811 nude 11 oban 11 23 29 + 911 953 odious 11 41 131 + 999998903 999999191 Ormiston 1931 34631 37379 + 1999987331 1999995713 palindromic 11 131 191 + 997909799 999212999 palprime 11 131 191 + 997909799 999212999 pancake 11 29 191 + 9948327041 9991960931 panconsummate 11 23 53 89 239 pandigital 11 partition 11 pernicious 11 41 131 + 9998903 9999299 Perrin 29 43721 plaindrome 11 23 29 + 6888899999 7777778999 prime 11 23 29 + 9999999701 10000000343 primeval 113 1013 1003679 100234679 Proth 41 113 641 + 9975365633 9996206081 repdigit 11 repunit 28792661 78914411 943280801 7294932341 self 53 233 659 + 999990581 999998903 self-describing 10311533 12313319 15333119 + 4419414113 4441411913 sliding 11 29 641 strobogrammatic 11 16091 1881881 + 6986889869 6988968869 strong prime 11 29 41 + 99998603 99999611 super-d 131 281 419 + 9998633 9998903 tetranacci 29 trimorphic 251 74218751 truncatable prime 23 29 53 + 9391564373 9439663853 twin 11 29 41 + 999998141 999999191 uban 11 23 29 + 9093000029 9096000029 Ulam 11 53 131 + 9986849 9991259 undulating 131 191 upside-down 1289 1559 7193 + 9992648111 9993557111 weak prime 23 83 89 + 99999551 99999563 weakly prime 3326489 5575259 7469789 + 9997470629 9999130709 Woodall 23 191 4373 590489 Zuckerman 11 zygodrome 11 44111 77711 + 9996661133 9999988811