Prime number sure

A prime number sure is a Prime number form 2 p + 1, where p is also a prime number. Reciprocally, the prime number p is a Prime number of Sophie Germain. The first prime numbers sure are:

5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019,1187,1283,1307,1319,1367,1439,1487,1523,1619,1823,1907

These prime numbers are called “sure” because of their application in the algorithms of Cryptologie such as the Algorithme of Diffie-Hellman. It must of course be noted that no prime number lower than 10⁵⁰ is really protected owing to the fact that any modern Ordinateur with an adapted algorithm can determine to them primality in a reasonable time. But the small numbers first sure are still very useful to learn the principles from these systems. There does not exist special Test of primality for the prime numbers sure as what exists for the prime numbers of Fermat and the prime numbers of Mersenne.

With share 5, there is no prime number of Fermat which is also a prime number sure. Indeed, if F is a prime number of Fermat, then ( F - 1) /2 is a power of two. To be first, this number must be equal to 2. Thus F =5.

With share 7, there is no prime number of Mersenne which is also a prime number sure. The demonstration is a little bit more complicated, but still in the field of the basic algebra. It should be known that p must be first so that 2^p - 1 can the being too. So that 2^p-1 is a prime number sure, it is necessary that the two numbers 2^p-1 and ((2^p - 1) - 1) /2 = 2^{p - 1} - 1 are numbers of Mersenne. Thus p and p -1 must be first both. Thus p =3 and 2^p - 1 = 7.

Like each term, except the last, of a Chaîne of Cunningham of first species is a prime number of Sophie Germain, therefore each term except the first of such a chain is a prime number sure. The prime numbers sure finishing by 7, of form 10 N + 7, are the last terms in such chains when they arrive, since 2 (10 N + 7) + 1 = 20 N + 15.

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