Prime number of Sophie Germain
A Prime number p is called a prime number of Sophie Germain if 2 p + 1 is also a prime number. They received a significance because of the demonstration of Sophie Germain in connection with the veracity of the Dernier theorem of Fermat for such prime numbers. It is conjectured that there exists an infinity of prime numbers of Sophie Germain, but, like the Conjecture of the prime numbers twins, this was not shown yet. The first small numbers first of Sophie Germain are:
- 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233,…
- (A005384 continuation in the electronic Encyclopédie of the whole continuations).
An estimate Heuristique for the quantity of prime numbers of Sophie Germain lower than N is 2 C 2 N / (ln N ) ² where C 2 is the constant prime numbers twins, roughly equal to 0,660161. For N = 104, this estimate predicts 156 prime numbers of Sophie Germain, who of 20% of error is compared with the exact value of 190 above. For N = 107, the estimate predicts 50.822, which is of a variation of 10% compared to the exact value of 56.032.
A continuation { p , 2 p + 1, 2 (2 p + 1) + 1,…} prime numbers of Sophie Germain is called a Chaîne of Cunningham of first species. Each term of such a continuation, except the first, is at the same time a prime number of Sophie Germain and a Prime number sure.