

A124112


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


3



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
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OFFSET

1,1


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 byproducts of the search for prime GaussianMersenne 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 GaussianMersenne (1+I)^k1 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 20062007 (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


LINKS



EXAMPLE

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.


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]] (* JeanFrançois Alcover, Feb 02 2015 *)


PROG

(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 */


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.


KEYWORD

nonn


AUTHOR

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


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


STATUS

approved



