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A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. They acquired significance because of Sophie Germain's proof that Fermat's last theorem is true for such primes. It is conjectured that there are infinitely many Sophie Germain primes, but like the twin prime conjecture, this has not been proven. The first few Sophie Germain primes are:

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113Integers 113 is the natural number following 112 and preceding 114. One hundred thirteen is the 30th prime number, following one hundred nine and preceding one hundred twenty-seven, a Sophie Germain prime and a permutable prime with 131 and 311. 113 is a, 131, 173, 179, 191, 233, ...

(sequence A005384 in OEIS).

A heuristicFor heuristics in computer science, see heuristic (computer science Heuristic is the art and science of discovery and invention. The word comes from the same Greek root (`ευρισκω) as " eureka," meaning "to find". estimate for the numberA number is an abstract entity used to describe quantity. There are different types of numbers. The most familiar numbers are the whole numbers {0, 1, 2,. denoted by W and the natural numbers {1, 2, 3,. used for counting and denoted by N . If the negative of Sophie Germain primes less than n is 2C₂ n / ( lnThe natural logarithm is the logarithm to the base e where e is approximately equal to 2. no exact fraction can be given, as e is an irrational number just like pi). The natural logarithm is defined for all positive real numbers x and can also be defined 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 above. For n = 10⁷, the estimate predicts 50822, whichs is still 10% off from the exact value of 56032.

A sequence {p, 2p + 1, 2(2p + 1) + 1, ...} of Sophie Germain primes is called a Cunningham chainAnalytic number theory In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. Cunningham. A Cunningham chain of the first kind is a sequence of prime numbers p . p such that for all 1 of the first kind. Every term of such a sequence except the first and last is both a Sophie Germain prime and a safe primeA safe prime is a prime number of the form 2''p + 1, where p is also a prime. Conversely, the prime p is a Sophie Germain prime. The first few safe primes are 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719,.