A227553 Number of solutions to x^2 - y^2 - z^2 == 1 (mod n).

1, 4, 6, 8, 30, 24, 42, 32, 54, 120, 110, 48, 182, 168, 180, 128, 306, 216, 342, 240, 252, 440, 506, 192, 750, 728, 486, 336, 870, 720, 930, 512, 660, 1224, 1260, 432, 1406, 1368, 1092, 960, 1722, 1008, 1806, 880, 1620, 2024, 2162, 768, 2058, 3000, 1836

Offset: 1

Author

José María Grau Ribas, Jul 16 2013

Keywords

• nonn
• mult

Comments

Conjecture: a(2) = 4; if s > 1 then a(2^s) = 2^(2s-1); if p == 1 (mod 4) then a(p^s) = (p+1)*p^(2s-1); if p == 3 (mod 4) then a(p^s) = (p-1)*p^(2s-1).

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..2500
• L. Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 14.11.6.

Crossrefs

Cf. A208895, A086932, A089003, A060968, A087784.

Programs

Mathematica

a[1] = 1; a[n_] := Sum[If[Mod[a^2-b^2-c^2, n] == 1, 1, 0], {a, n}, {b, n}, {c, n}]; Table[a[n], {n, 10}]

PARI

M(n,f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))}
a(n)={polcoeff(lift(M(n, i->i^2) * M(n, i->-(i^2))^2 ), 1%n)} \\ Andrew Howroyd, Jun 24 2018