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A178444 Markov numbers that are prime. 6
2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 1686049, 2922509, 3276509, 94418953, 321534781, 433494437, 780291637, 1405695061, 2971215073, 19577194573, 25209506681, 44208781349, 44560482149, 128367472469 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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]

All terms after the first are of the form 4k+1. [Paul Muljadi, Jan 31 2011]

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..300

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

MATHEMATICA

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]

CROSSREFS

Cf. A002559, A256395.

Sequence in context: A045703 A289843 A242080 * A299145 A122025 A236414

Adjacent sequences:  A178441 A178442 A178443 * A178445 A178446 A178447

KEYWORD

nonn

AUTHOR

Paul Muljadi, Jan 01 2011

STATUS

approved

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Last modified April 7 19:17 EDT 2020. Contains 333306 sequences. (Running on oeis4.)