A124112 Numbers n such that ((1+I)^n+1)/(2+I) is a Gaussian prime.

5, 7, 9, 11, 13, 17, 29, 43, 53, 89, 283, 557, 563, 613, 691, 1223, 2731, 5147, 5323, 5479, 9533, 10771, 11257, 11519, 12583, 23081, 36479, 52567, 52919, 125929, 221891, 235099, 305867, 311027, 333227, 365689, 792061, 1127239, 1148729, 1347781, 1669219, 1882787, 2305781, 4533073, 5243339

Offset: 1

Author

David J. Broadhurst and Jean Penne (jpenne(AT)wanadoo.fr), Nov 27 2006

Keywords

• nonn

Comments

These numbers have been proved prime only up to exponent a(25) = 12583.

With the only exception of a(3) = 9, it is easy to prove that ((1+I)^a(n)+1)/(2+I) prime => a(n) prime. Following an idea of Harsh Aggarwal, many of these numbers have been discovered as by-products of the search for prime Gaussian-Mersenne norms. The reason for this is the Aurifeuillan factorization of M(k) = 2^(2k) + 1 with k odd. These numbers can be written as M(k) = GM(k)*GQ(k)*5 where GM(k) is the norm of the Gaussian-Mersenne (1+I)^k-1 while GQ(k) is the norm of ((1+I)^a(n)+1)/(2+I). This allowed us to write a program which can simultaneously prove the primality of GM(k) and, without extra cost, the probable primality of GQ(k). Using this program, Borys Jaworski (discoverer of the presently largest known GM) also discovered an outlier of this sequence: a(?) = 1127239.

The terms 1127239 and 1148729 were found by Borys Jaworski in 2006-2007 (see PRP Records link). These two terms also belong to A124165(n) = Primes p such that (2^p + 2^((p+1)/2) + 1)/5 is prime. a(n) is a union of the only composite term a(3) = 9 and two prime sequences: A124165(n) and A125742(n) = Primes p such that (2^p - 2^((p+1)/2) + 1)/5 is prime. - Alexander Adamchuk, Jun 20 2007

The term 12503723 is also in the sequence but its position is unknown. - Serge Batalov, Jul 17 2020

Examples

			For n = 27, ((1+I)^36479+1)/(2+I) is a probable Gaussian prime because its norm, (2^36479+2^18240+1)/5, is a Fermat PRP.
		

Links

• Henri Lifchitz & Renaud Lifchitz, PRP Records. Probable Primes Top 10000.

Crossrefs

Cf. A124165 = Primes p such that (2^p + 2^((p+1)/2) + 1)/5 is prime.

Cf. A125742 = Primes p such that (2^p - 2^((p+1)/2) + 1)/5 is prime.

Programs

Mathematica

(* A naive script not convenient for large terms *) Reap[For[n = 2, n < 10^4, n = If[n == 7, 9, NextPrime[n]], If[PrimeQ[((1 + I)^n + 1)/(2 + I), GaussianIntegers -> True], Print[n]; Sow[n]] ]][[2, 1]] (* Jean-François Alcover, Feb 02 2015 *)

PARI

forprime(n=3, 2731, if(ispseudoprime((2^n+kronecker(2, n)*2^((n+1)/2)+1)/5), print1(n ", "))); /* Serge Batalov, Mar 31 2014 */

Extensions

a(37) from Thomas Ritschel (see PRP Records). - Serge Batalov, Mar 31 2014

a(38)-a(42) from Borys Jaworski (see PRP Records). - Serge Batalov, Mar 31 2014

a(43)-a(44) from Serge Batalov, Mar 31 2014

a(45) from Serge Batalov, Jul 17 2020