cp's OEIS Frontend

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

%I A124112 #22 Jul 18 2020 02:11:50
%S A124112 5,7,9,11,13,17,29,43,53,89,283,557,563,613,691,1223,2731,5147,5323,
%T A124112 5479,9533,10771,11257,11519,12583,23081,36479,52567,52919,125929,
%U A124112 221891,235099,305867,311027,333227,365689,792061,1127239,1148729,1347781,1669219,1882787,2305781,4533073,5243339
%N A124112 Numbers n such that ((1+I)^n+1)/(2+I) is a Gaussian prime.
%C A124112 These numbers have been proved prime only up to exponent a(25) = 12583.
%C A124112 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.
%C A124112 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
%C A124112 The term 12503723 is also in the sequence but its position is unknown. - _Serge Batalov_, Jul 17 2020
%H A124112 Henri Lifchitz & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%282^%3F2^%3F%2B1%29%2F5&amp;action=Search">PRP Records. Probable Primes Top 10000</a>.
%e A124112 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.
%t A124112 (* 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 *)
%o A124112 (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 */
%Y A124112 Cf. A124165 = Primes p such that (2^p + 2^((p+1)/2) + 1)/5 is prime.
%Y A124112 Cf. A125742 = Primes p such that (2^p - 2^((p+1)/2) + 1)/5 is prime.
%K A124112 nonn
%O A124112 1,1
%A A124112 David J. Broadhurst and Jean Penne (jpenne(AT)wanadoo.fr), Nov 27 2006
%E A124112 a(37) from Thomas Ritschel (see PRP Records). - _Serge Batalov_, Mar 31 2014
%E A124112 a(38)-a(42) from Borys Jaworski (see PRP Records). - _Serge Batalov_, Mar 31 2014
%E A124112 a(43)-a(44) from _Serge Batalov_, Mar 31 2014
%E A124112 a(45) from _Serge Batalov_, Jul 17 2020