A256395 Composite Markoff numbers.

34, 169, 194, 610, 985, 1325, 4181, 6466, 9077, 10946, 14701, 37666, 51641, 62210, 75025, 135137, 195025, 196418, 294685, 499393, 646018, 925765, 1136689, 1278818, 1346269, 1441889, 2012674, 2423525, 3524578, 4400489, 6625109, 7453378, 8399329, 9227465, 9647009

Offset

1,1

Comments

Bourgain, Gamburd, and Sarnak have announced a proof that almost all Markoff numbers A002559 are composite. Equivalently, the prime Markoff numbers A178444 have density zero among all Markoff numbers. (It is conjectured that infinitely many Markoff numbers are prime.)

See A002559 for references, links, and additional comments.

Links

Table of n, a(n) for n=1..35.

J. Bourgain, A. Gamburd, and P. Sarnak, Markoff triples and strong approximation, arXiv:1505.06411 [math.NT], 2015.

Mathematica

Rest[Select[m = {1};
  Do[x = m[[i]]; y = m[[j]]; a = (3*x*y + Sqrt[-4*x^2 - 4*y^2 + 9*x^2*y^2])/2;
    b = (3*x*y + Sqrt[-4*x^2 - 4*y^2 + 9*x^2*y^2])/2;
   If[IntegerQ[a], m = Union[Join[m, {a}]]];
   If[IntegerQ[b], m = Union[Join[m, {b}]]], {n, 8}, {i, Length[m]}, {j, i}];
  Take[m, 50], ! PrimeQ[#] &]]

Prog

(SageMath)
def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:
    # Using function 'MarkovNumbers' from A002559.
    M = MarkovNumbers(len, MAX)
    U = set([])
    for m in M:
        if not is_prime(ZZ(m)):
            U.add(m)
    return sorted(U)[1:len]
# Balance required sequence length and search depth.
print(A386894List(len=56))  # Peter Luschny, Aug 12 2025

Crossrefs

Intersection of A002808 and A002559.

Complement of (A178444 union {1}) in A002559.

Sequence in context: A345126 A212407 A190607 * A182585 A191593 A259944

Adjacent sequences: A256392 A256393 A256394 * A256396 A256397 A256398

Keyword

nonn

Author

Jonathan Sondow, Apr 30 2015

Status

approved