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A270697 Composite numbers k == 3 (mod 4) such that (1 + i)^k == 1 - i (mod k), where i = sqrt(-1).

%I A270697 #58 Nov 06 2023 00:26:35
%S A270697 2047,42799,90751,256999,271951,476971,514447,741751,877099,916327,
%T A270697 1302451,1325843,1397419,1441091,1507963,1530787,1907851,2004403,
%U A270697 2205967,2304167,2748023,2811271,2953711,2976487,3090091,3116107,4469471,4863127,5016191
%N A270697 Composite numbers k == 3 (mod 4) such that (1 + i)^k == 1 - i (mod k), where i = sqrt(-1).
%C A270697 Composite k == 3 (mod 4) such that 2*(-4)^((k-3)/4) == -1 (mod k). - _Robert Israel_, Mar 21 2016
%C A270697 2*(-4)^((p-3)/4) == -1 (mod p) is satisfied by all primes p == 3 (mod 4), see A318908. - _Jianing Song_, Sep 05 2018
%C A270697 Numbers in A047713 that are congruent to 3 mod 4. Most terms are congruent to 7 mod 8. For terms congruent to 3 mod 8, see A244628. - _Jianing Song_, Sep 05 2018
%C A270697 Question: Is this a subsequence of A001262? I have verified that it contains all terms up to 2^64. - _Joseph M. Shunia_, Jul 02 2019
%H A270697 Amiram Eldar, <a href="/A270697/b270697.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..111 from Jianing Song using data from A047713)
%p A270697 select(t -> not isprime(t) and 1 + 2*(-4) &^ ((t-3)/4) mod t = 0, [seq(i, i=7..10^7, 4)]); # _Robert Israel_, Mar 21 2016
%t A270697 Select[3 + 4*Range[10000000], PrimeQ[#] == False && PowerMod[1 + I, #, #] == Mod[1 - I, #] &]
%o A270697 (PARI) forstep(n=3, 10^7, 4, if(Mod(2, n)^((n-1)/2)==kronecker(2, n) && !isprime(n), print1(n, ", ")))
%Y A270697 Subsequence of A001567 and A047713.
%Y A270697 A244628 is a proper subsequence.
%Y A270697 Cf. A001262, A006971, A270698, A318908.
%K A270697 nonn
%O A270697 1,1
%A A270697 _José María Grau Ribas_, Mar 21 2016