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A385188 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is not divisible by 2 or 3.

%I A385188 #14 Jun 22 2025 21:32:26
%S A385188 599,691,1291,1451,2411,3851,4919,5051,5479,5531,5879,6599,7079,7691,
%T A385188 8011,8039,11491,13291,14011,15091,15971,16651,17359,18731,19211,
%U A385188 19531,20731,22651,23971,24611,25639,25679,26251,32051,32359,32531,32771,32971,35879,37039,37571,38011,38371
%N A385188 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is not divisible by 2 or 3.
%C A385188 Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
%C A385188 5 is a quadratic residue of integers modulo p for p being a term of this sequence. (See A385165).
%H A385188 Jianing Song, <a href="/A385188/b385188.txt">Table of n, a(n) for n = 1..10000</a>
%e A385188 5479 is a term since (2+-i)^125081 == 1 (mod 5479), and 125081 is divisible by neither 2 nor 3.
%o A385188 (PARI) ord(p) = my(d = divisors((p+1)*znorder(Mod(5, p)))); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i, p)
%o A385188 isA385188(p) = isprime(p) && p%4==3 && ord(p)%2 && ord(p)%3
%Y A385188 Cf. A385165, A385179, A385219 (the actual multiplicative orders).
%Y A385188 this sequence < A385169 < A385180 < A385167 < intersection of A122869 and A385168, where Ax < Ay means that Ax is a subsequence of Ay.
%Y A385188 Also a subsequence of A385191.
%K A385188 nonn,easy
%O A385188 1,1
%A A385188 _Jianing Song_, Jun 20 2025