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%I #45 Dec 13 2023 08:17:52
%S 2,5,13,29,89,233,433,1597,2897,5741,7561,28657,33461,43261,96557,
%T 426389,514229,1686049,2922509,3276509,94418953,321534781,433494437,
%U 780291637,1405695061,2971215073,19577194573,25209506681,44208781349,44560482149,128367472469
%N Markov numbers that are prime.
%C Triples of prime Markov numbers appear to be very rare. For Markov numbers less than 10^1000, only five are known: (2, 5, 29), (5, 29, 433), (5, 2897, 43261), (2, 5741, 33461), and (89, 6017226864647074440629, 1606577036114427599277221). Note that the smallest members of these triples are prime Fibonacci numbers 2, 5, and 89. [_T. D. Noe_, Jan 28 2011]
%C All terms after the first are of the form 4k+1. [_Paul Muljadi_, Jan 31 2011]
%C Bourgain, Gamburd, and Sarnak have announced a proof that almost all Markoff numbers are composite--see A256395. Equivalently, the present sequence has density zero among all Markoff numbers. (It is conjectured that the sequence is infinite.) - _Jonathan Sondow_, Apr 30 2015
%H T. D. Noe, <a href="/A178444/b178444.txt">Table of n, a(n) for n = 1..300</a>
%H Jean Bourgain, Alex Gamburd, and Peter Sarnak, <a href="https://arxiv.org/abs/1505.06411">Markoff Triples and Strong Approximation</a>, arXiv:1505.06411 [math.NT], 2015.
%H Yasuaki Gyoda and Shuhei Maruyama, <a href="https://arxiv.org/abs/2312.07329">Uniqueness theorem of generalized Markov numbers that are prime powers</a>, arXiv:2312.07329 [math.NT], 2023. See Appendix A.
%H Kristin DeVleming and Nikita Singh, <a href="https://arxiv.org/abs/2311.15922">Rational unicuspidal plane curves of low degree</a>, arXiv:2311.15922 [math.AG], 2023. See p. 14.
%t 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, 40] (* Robert G. Wilson v, Oct 05 2005, taken from A002559 *); Select[m, PrimeQ]
%Y Cf. A002559, A256395.
%K nonn
%O 1,1
%A _Paul Muljadi_, Jan 01 2011
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