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Composite Markoff numbers.
3

%I #15 May 29 2015 14:04:51

%S 34,169,194,610,985,1325,4181,6466,9077,10946,14701,37666,51641,62210,

%T 75025,135137,195025,196418,294685,499393,646018,925765,1136689,

%U 1278818,1346269,1441889,2012674,2423525,3524578

%N Composite Markoff numbers.

%C 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 See A002559 for references, links, and additional comments.

%H 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 Rest[Select[m = {1};

%t 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 b = (3*x*y + Sqrt[-4*x^2 - 4*y^2 + 9*x^2*y^2])/2;

%t If[IntegerQ[a], m = Union[Join[m, {a}]]];

%t If[IntegerQ[b], m = Union[Join[m, {b}]]], {n, 8}, {i, Length[m]}, {j, i}];

%t Take[m, 50], ! PrimeQ[#] &]]

%Y Intersection of A002808 and A002559.

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

%K nonn,more

%O 1,1

%A _Jonathan Sondow_, Apr 30 2015