A190637 Primes p == 3 mod 4 whose index as prime divides their index as a Gaussian prime (in the first quadrant, as defined in A103431, for example).

3, 43, 7639, 25703, 38371, 61291

Offset: 1

Author

Sven Simon, May 15 2011

Keywords

• nonn
• more

Comments

The index of a prime p = 3 mod 4 as a Gaussian prime is well defined, it is summed up by 1 for the complex prime 1+i (as factor of prime 2 = -i*(1+i)^2).

The count of primes (3 mod 4) <= p, which remain unchanged as they cannot be factored further into complex primes 2 times the count of primes (1 mod 4) <= p**2 (such primes p1 are split into two distinct complex primes of the first quadrant with size sqrt(p1)).

As the result from the splitting of the primes 1 mod 4, the indices of primes 3 mod 4 as Gaussian prime grows rather rapidly against their index as normal prime.

Interesting numerical effects: the prime index of 43 is 14, with 3*14+1 = 43. 43 is the upper part of twin prime with 41 (which would be 14*3 - 1 with an index 14, if 1 was counted as prime). 4241 and 4243 are both primes.

The ratio f between both indices can be estimated as f = (p^2 / log(p^2)) / (p / log(p)) = p/2. - Sven Simon, May 26 2011

Examples

			The prime 3 has index 2, as a Gaussian prime it has index 4 (the list is 1+i, 1+2i, 2i+1, 3, ...), and 2 divides 4.
		

Crossrefs

Cf. A103431 (Gaussian primes in first quadrant), A190634 (prime index), A190635 (index as Gaussian prime).

Extensions

Changed name definition which was a bit wrong, the index is not a prime number