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A256395 Composite Markoff numbers.

%I A256395 #20 Aug 12 2025 14:55:27
%S A256395 34,169,194,610,985,1325,4181,6466,9077,10946,14701,37666,51641,62210,
%T A256395 75025,135137,195025,196418,294685,499393,646018,925765,1136689,
%U A256395 1278818,1346269,1441889,2012674,2423525,3524578,4400489,6625109,7453378,8399329,9227465,9647009
%N A256395 Composite Markoff numbers.
%C A256395 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.)
%C A256395 See A002559 for references, links, and additional comments.
%H A256395 J. Bourgain, A. Gamburd, and P. Sarnak, <a href="http://arxiv.org/abs/1505.06411">Markoff triples and strong approximation</a>, arXiv:1505.06411 [math.NT], 2015.
%t A256395 Rest[Select[m = {1};
%t A256395   Do[x = m[[i]]; y = m[[j]]; a = (3*x*y + Sqrt[-4*x^2 - 4*y^2 + 9*x^2*y^2])/2;
%t A256395     b = (3*x*y + Sqrt[-4*x^2 - 4*y^2 + 9*x^2*y^2])/2;
%t A256395    If[IntegerQ[a], m = Union[Join[m, {a}]]];
%t A256395    If[IntegerQ[b], m = Union[Join[m, {b}]]], {n, 8}, {i, Length[m]}, {j, i}];
%t A256395   Take[m, 50], ! PrimeQ[#] &]]
%o A256395 (SageMath)
%o A256395 def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:
%o A256395     # Using function 'MarkovNumbers' from A002559.
%o A256395     M = MarkovNumbers(len, MAX)
%o A256395     U = set([])
%o A256395     for m in M:
%o A256395         if not is_prime(ZZ(m)):
%o A256395             U.add(m)
%o A256395     return sorted(U)[1:len]
%o A256395 # Balance required sequence length and search depth.
%o A256395 print(A386894List(len=56))  # _Peter Luschny_, Aug 12 2025
%Y A256395 Intersection of A002808 and A002559.
%Y A256395 Complement of (A178444 union {1}) in A002559.
%K A256395 nonn
%O A256395 1,1
%A A256395 _Jonathan Sondow_, Apr 30 2015