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

Author

Jonathan Sondow, Apr 30 2015

Keywords

• nonn

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

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

Crossrefs

Intersection of A002808 and A002559.

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

Programs

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[#] &]]

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