This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385190 #12 Jun 21 2025 00:45:53
%S A385190 31,127,191,223,383,479,863,1151,1439,1471,1823,2111,2143,2207,2399,
%T A385190 2591,2687,2879,3167,3359,3391,4127,4703,4799,5087,5279,5471,5503,
%U A385190 6047,6079,6143,6271,6719,6911,7103,7487,7583,8191,8287,8447,8543,8831,8863,9311,9439,9631,9791,9887
%N A385190 Primes p == 3 (mod 4), p > 3 such that 1+-i are 24th powers modulo p.
%C A385190 Note that the primes congruent to 3 modulo 4 are precisely the rational primes in the ring of Gaussian integers.
%C A385190 Primes p == 3 (mod 4), p > 3 such that (1+-i)^((p^2-1)/24) == 1 (mod p). Note that p^2-1 is always divisible by 24 for primes p > 3.
%C A385190 Primes p = A002145(k) such that the multiplicative order of 1+-i modulo p (A385163(k)) divides (p^2-1)/24. Since A385165(k) = 4*ord(-4,p), this is also primes p == 3 (mod 4) such that 96*ord(-4,p) divides p^2-1, where ord(a,p) is the multiplicative order of a modulo p.
%C A385190 Sequence is infinite since it contains all primes congruent to 95 modulo 96.
%C A385190 Primes p == 3 (mod 4), p > 3 such that [1,-1;1,1]^((p^2-1)/24) or [1,1;-1,1]^((p^2-1)/24) == I_2 (mod p).
%C A385190 Since 96 divides p^2-1 for p being a term of this sequence, we must have p == 15 (mod 16).
%H A385190 Jianing Song, <a href="/A385190/b385190.txt">Table of n, a(n) for n = 1..10000</a>
%e A385190 31 is a term since (1+-i)^((31^2-1)/24) = (-4)^((31^2-1)/96) = 1048576 == 1 (mod 31). Indeed, the solutions to x^24 == 1+i (mod 31) are x == {17-6*i, 16+6*i, 1+8*i, -1+13*i, 9-5*i, 3+5*i} X {+-1, +-i} (mod 31).
%o A385190 (PARI) isA385190(p) = isprime(p) && p%16==15 && Mod(-4,p)^((p^2-1)/96) == 1
%Y A385190 Cf. A385163, A385191 (2+-i are 24th powers), A002145.
%K A385190 nonn,easy
%O A385190 1,1
%A A385190 _Jianing Song_, Jun 20 2025