A329705 Composite numbers k such that (1 - w)^(k-1) == 1 (mod k) in the ring of Eisenstein integers (w = (-1 + sqrt(3)*i)/2).

121, 703, 1729, 1891, 2821, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009, 29341, 31621, 41041, 44287, 46657, 47197, 49141, 50881, 52633, 55969, 63139, 63973, 74593, 75361, 79003, 82513, 87913, 88573, 93961, 97567, 105163

Offset: 1

Author

Amiram Eldar, Feb 28 2020

Keywords

• nonn

Comments

w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2, where i is the imaginary unit, is a unit in the ring of Eisenstein integers (usually denoted by the Greek letter omega).

Also Euler-Jacobi pseudoprimes to base 3 that are congruent to 1 (mod 6).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weissteins's World of Mathematics, Eisenstein Integer.
• Wikipedia, Eisenstein integer.

Crossrefs

Intersection of A016921 and A048950.

Cf. A066408, A270698.

Programs

Mathematica

eisProd[z1_, z2_] := {z1[[1]]*z2[[1]] - z1[[2]]*z2[[2]], z1[[1]]*z2[[2]] + z1[[2]]*z2[[1]] - z1[[2]]*z2[[2]]}; seq = {}; z = {1, 0}; Do[z = eisProd[{1, -1}, z]; If[CompositeQ[n] && And @@ Divisible[z - {1, 0}, n], AppendTo[seq, n]], {n, 2, 10^4}]; seq