A227867 Number of Lipschitz quaternions X such that X^2 == 1 (mod n).

1, 8, 14, 32, 32, 112, 58, 32, 110, 256, 134, 448, 184, 464, 448, 32, 308, 880, 382, 1024, 812, 1072, 554, 448, 752, 1472, 974, 1856, 872, 3584, 994, 32, 1876, 2464, 1856, 3520, 1408, 3056, 2576, 1024, 1724, 6496, 1894, 4288, 3520, 4432, 2258, 448, 2746, 6016, 4312, 5888

Offset: 1

Author

José María Grau Ribas, Nov 02 2013

Keywords

• nonn
• mult

Comments

A quaternion q = a + bi + cj + dk is congruent to 1 (mod n) iff a == 1 (mod n) and b == c == d == 0 (mod n).

Links

• Wikipedia, Hurwitz quaternion.

Crossrefs

Cf. A227477, A227499, A227628.

Programs

Mathematica

cuaternios[n_] := Flatten[Table[{{a, -b, d, -c}, {b, a, -c, -d}, {-d, c, a, -b}, {c, d, b, a}}, {a, n}, {b, n}, {c, n}, {d, n}], 3]; invo[n_] := invo[n] = Length@Select[cuaternios[n], Mod[#.# - IdentityMatrix[4],n] == 0*# &]; Table[invo[n], {n, 1, 25}]

Extensions

More terms from Amiram Eldar, May 06 2024