A145918 Subset of Sophie Germain primes generated by an iterative process starting from 11.

11, 23, 83, 179, 359, 719, 1439, 2903, 5903, 11831, 23819, 47639, 95723, 191459, 383219, 766763, 1533599, 3067511, 6135023, 12271019, 24542351, 49085819, 98172131, 196344719, 392689439, 785379359, 1570758719, 3141519443

Offset: 1

Author

Reikku Kulon, Oct 24 2008

Keywords

• easy
• nice
• nonn

Comments

Prime numbers of this form are exceptionally easy to generate. The hundredth term in this sequence is 14835409839824806270695281050319, which can be determined to be prime in a fraction of a second, or produced starting from 11 in less than a second using a 2GHz Athlon. The number of decimal digits increases by one within four iterations.

The time needed grows according to O(n^log(4)) for iterations n, or O(log(v)^log(4)) for values v.

Note that these are considered to be safe primes for cryptography. This may be very unwise, as the average time to produce any particular value is many orders of magnitude less than its size. Consider that the guaranteed prime 4.149...063 * 10^278 can be generated in fifteen seconds.

Also note the surprising value of the final term given.

Links

• C. K. Caldwell, The Prime Glossary, Sophie Germain Prime.
• C. K. Caldwell, Cunningham Chain.
• Eric Weisstein's World of Mathematics, Figurate Number.

Crossrefs

Cf. A005384, A005385 (Sophie Germain primes).

Cf. A000225, A006516, A000217.

Formula

Define:

n, a positive integer congruent to 11 (mod 12);

o, 2n + 1;

Mn, Mersenne number n: A000225(n);

Pn, pseudoperfect number n: A006516(n) and note its simple construction from Mn: Pn = A000217(A000225(n)) = (Mn^2 + Mn + 1) / 2 = (4^n - 2^n) / 2;

Fo, figurate kernel o = A000217(o) - o = (o^2 - o) / 2.

Observe that Pn (mod Fo) is calculable by modular exponentiation.

Then n is a Sophie Germain prime and o is its matching safe prime iff Pn is congruent to o (mod Fo). n and o are therefore members of a Cunningham chain.