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

2047, 42799, 90751, 256999, 271951, 476971, 514447, 741751, 877099, 916327, 1302451, 1325843, 1397419, 1441091, 1507963, 1530787, 1907851, 2004403, 2205967, 2304167, 2748023, 2811271, 2953711, 2976487, 3090091, 3116107, 4469471, 4863127, 5016191

Offset: 1

Author

José María Grau Ribas, Mar 21 2016

Keywords

• nonn

Comments

Composite k == 3 (mod 4) such that 2*(-4)^((k-3)/4) == -1 (mod k). - Robert Israel, Mar 21 2016

2*(-4)^((p-3)/4) == -1 (mod p) is satisfied by all primes p == 3 (mod 4), see A318908. - Jianing Song, Sep 05 2018

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

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..111 from Jianing Song using data from A047713)

Crossrefs

Subsequence of A001567 and A047713.

A244628 is a proper subsequence.

Cf. A001262, A006971, A270698, A318908.

Programs

Maple

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

Mathematica

Select[3 + 4*Range[10000000], PrimeQ[#] == False && PowerMod[1 + I, #, #] == Mod[1 - I, #] &]

PARI

forstep(n=3, 10^7, 4, if(Mod(2, n)^((n-1)/2)==kronecker(2, n) && !isprime(n), print1(n, ", ")))