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A385169 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is odd.

%I A385169 #27 Jun 26 2025 14:31:02
%S A385169 331,571,599,691,839,971,1051,1171,1291,1451,1571,1879,2131,2411,2971,
%T A385169 3251,3331,3491,3571,3691,3851,4051,4091,4211,4651,4679,4691,4919,
%U A385169 4931,5051,5171,5479,5531,5651,5839,5851,5879,6011,6599,6679,6691,7079,7211,7331,7691,8011,8039,8171,8731,8839,9011,9371,9811
%N A385169 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is odd.
%C A385169 Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
%C A385169 Let ord(a,m) be the multiplicative order of a modulo m. (Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i]). For a prime p == 3 (mod 4), we have that ord(2+-i,p) is divisible by ord(5,p), and that ord(2+-i,p) divides (p+1) * ord(5,p). What's more, ord(2+-i,p) divides (p^2-1)/2 if and only if 5 is a quadratic residue of integers modulo p. (See A385165).
%C A385169 As a result, if ord(2+-i,p) is not divisible by 8, then ord(5,p) is odd:
%C A385169   - Of course this is true if ord(2+-i,p) is odd.
%C A385169   - If ord(2+-i,p) == 2 (mod 4) and ord(5,p) is even, then ord(2+-i,p)/ord(5,p) is odd, and so ord(2+-i,p) divides ((p+1)/4) * ord(5,p), then ord(5,p) is odd. This implies that ord(2+-i,p) is odd, a contradiction.
%C A385169   - If ord(2+-i,p) == 4 (mod 8) and ord(5,p) is even (we have ord(5,p) == 2 (mod 4) since p == 3 (mod 4)), then ord(2+-i,p)/ord(5,p) == 2 (mod 4), and so ord(2+-i,p) divides ((p+1)/2) * ord(5,p), then ord(5,p) is odd. This implies that ord(2+-i,p) == 2 (mod 4), a contradiction.
%C A385169 From the above paragraph, this sequence is also primes p == 3 (mod 4) such that ord(2+-i,p)/ord(5,p) is odd.
%H A385169 Jianing Song, <a href="/A385169/b385169.txt">Table of n, a(n) for n = 1..10000</a>
%e A385169 8731 is a term since (2+-i)^635253 == 1 (mod 8731), and 635253 is odd.
%e A385169 8839 is a term since (2+-i)^57447 == 1 (mod 8839), and 57447 is odd.
%e A385169 9011 is a term since (2+-i)^2029953 == 1 (mod 9011), and 2029953 is odd.
%o A385169 (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 A385169 isA385169(p) = isprime(p) && p%4==3 && ord(p)%2
%Y A385169 Cf. A385165, A385179, A385192, A385217 (the actual multiplicative orders).
%Y A385169 A385188 < this sequence < A385180 < A385167 < intersection of A122869 and A385168, where Ax < Ay means that Ax is a subsequence of Ay.
%K A385169 nonn,easy
%O A385169 1,1
%A A385169 _Jianing Song_, Jun 20 2025