A385118 Number of solutions to x^2 + y^2 + z^2 == 3*x*y*z (mod n).

1, 5, 9, 24, 41, 45, 29, 96, 99, 205, 89, 216, 209, 145, 369, 320, 341, 495, 305, 984, 261, 445, 461, 864, 1125, 1045, 891, 696, 929, 1845, 869, 1280, 801, 1705, 1189, 2376, 1481, 1525, 1881, 3936, 1805, 1305, 1721, 2136, 4059, 2305, 2069, 2880, 1715, 5625

Offset

1,2

Comments

Conjecture: Every solution mod p lifts to an integer solution. See linked articles by Bourgain et al.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..125

Jean Bourgain, Alexander Gamburd, and Peter Sarnak, Markoff triples and strong approximation, arXiv:1505.06411 [math.NT], 2015.

Jean Bourgain, Alexander Gamburd, and Peter Sarnak, Markoff Surfaces and Strong Approximation: 1, arXiv:1607.01530 [math.NT], 2016.

Leonard Carlitz, The number of points on certain cubic surfaces over a finite field, Bollettino dell'Unione Matematica Italiana, Serie 3, Vol. 12 (1957), n.1, p. 19-21.

Mathematics Stack Exchange, Number of solutions to Markov Diophantine equation mod p.

Wikipedia, Markov number.

Formula

a(p) = p^2 + 3*p * Legendre(-1, p) + 1 for p prime, p > 3. (See linked post from Mathematics Stack Exchange.)

Mathematica

a[n_]:= Total[Boole[Mod[#1^2+#2^2+#3^2-3 #1 #2 #3, n]==0]&@@@Tuples[Range[0, n-1], 3]]; Table[a[n], {n, 1, 50}] (* Vincenzo Librandi, Nov 07 2025 *)

Prog

(Python)
from itertools import product
def a(n): return sum((x*x+y*y+z*z-3*x*y*z)%n == 0 for x, y, z in product(range(n), repeat=3))
print([a(n) for n in range(1, 51)])

Crossrefs

Sequence in context: A272256 A132354 A146419 * A365070 A387245 A162907

Adjacent sequences: A385115 A385116 A385117 * A385119 A385120 A385121

Keyword

nonn,mult

Author

David Radcliffe, Jun 18 2025

Status

approved