Sophie Germain prime
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.
A Sophie Germain prime p > 3 is of the form 6k−1 or, equivalently, p ≡ 5 (mod 6) — as is its matching safe prime 2p+1. We note that the other form for a prime p > 3 is 6k + 1 or, equivalently, p ≡ 1 (mod 6), and that 3|(2p + 1) — thus excluding such p from the Sophie Germain prime domain. This is trivially proven using modular arithmetic.
It is conjectured that there are infinitely many Sophie Germain primes, but this has not been proven.
| Are there infinitely many Sophie Germain primes? |
The first few Sophie Germain primes are:
The largest known Sophie Germain prime as of April 2012[update] is 18543637900515 × 2666667−1. It has 200701 decimal digits and was found in April 2012 by Philipp Bliedung in a distributed PrimeGrid search using the programs TwinGen and LLR.[1] The previous record was 183027 × 2265440−1 with 79911 digits, found in March 2010 by Tom Wu using LLR.[2] Before that the two largest were 648621027630345 × 2253824−1 and 620366307356565 × 2253824−1. They both have 76424 digits and were found in November 2009 by Zoltán Járai, Gabor Farkas, Timea Csajbok, János Kasza and Antal Járai.[3][4] The previous record was set 6 weeks earlier, 607095 × 2176311−1 with 53081 digits, found by Tom Wu.[5] Before that the record was 48047305725 × 2172403−1 with 51910 digits, found by David Underbakke in January 2007 using TwinGen and LLR.[6] And before that, the record was held by the same team as the November 2009 records, 137211941292195 × 2171960−1 with 51780 digits, found in May 2006.[7] As of April 2012[update] the above are still the seven largest known Sophie Germain primes.
A heuristic estimate (due to G. H. Hardy and J. E. Littlewood) for the number of Sophie Germain primes less than n is 2C2 n / (ln n)2 where C2 is the twin prime constant, approximately 0.660161. For n = 104, this estimate predicts 156 Sophie Germain primes, which has a 20% error compared to the exact value of 190. For n = 107, the estimate predicts 50822, which is still 10% off from the exact value of 56032.
A sequence {p, 2p + 1, 2(2p + 1) + 1, ...} of one or more Sophie Germain primes, ending with a prime which does not have to be a Sophie Germain, is called a Cunningham chain of the first kind. Every term of such a sequence except the first and last is both a Sophie Germain prime and a safe prime.
If a Sophie Germain prime p is congruent to 3 (mod 4), then its matching safe prime 2p + 1 will be a divisor of the Mersenne number 2p − 1.
Application in (pseudo-)random number generation [edit]
Sophie Germain primes have a practical application in the generation of pseudo-random numbers. The decimal expansion of 1/q will produce a stream of q − 1 pseudo-random digits, if q is the safe prime of a Sophie Germain prime p, with p congruent to 3, 9, or 11 (mod 20).[8] Thus “suitable” prime numbers q are 7, 23, 47, 59, 167, 179, etc. (corresponding to p = 3, 11, 23, 29, 83, 89, etc.). The result is a stream of length q − 1 digits (including leading zeros). So, for example, using q = 23 generates the pseudo-random digits 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3. Note that these digits are not appropriate for cryptographic purposes, as the value of each can be derived from its predecessor in the digit-stream.
In popular culture [edit]
Sophie Germain primes are mentioned in the stage play Proof and the subsequent film.
See also [edit]
- Lucas–Lehmer–Riesel test
- PrimeGrid – search for Sophie Germain primes
- Twin Prime Search – includes search for Sophie Germain primes
References [edit]
- ^ "PrimeGrid’s Sophie Germain Prime Search" (PDF). PrimeGrid. Retrieved 18 April 2012.
- ^ The Prime Database: 183027*2^265440-1. From The Prime Pages.
- ^ The Prime Database: 648621027630345*2^253824-1.
- ^ The Prime Database: 620366307356565*2^253824-1
- ^ The Prime Database: 607095*2^176311-1.
- ^ The Prime Database: 48047305725*2^172403-1.
- ^ The Prime Database: 137211941292195*2^171960-1.
- ^ Matthews, R. A. J. (1992), "Maximally Periodic Reciprocals", Bulletin of the Institute of Mathematics and its Applications 28: 147–148
External links [edit]
- The Top Twenty Sophie Germain Primes — from the Prime Pages.
- "Sloane's A000355 : Primes = 3, 9, 11 (mod 20) such that 2p+1 is also prime", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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