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
Jean Bourgain, Alex Gamburd, and Peter Sarnak, Markoff Triples and Strong Approximation, arXiv:1505.06411 [math.NT], 2015.
Yasuaki Gyoda and Shuhei Maruyama, Uniqueness theorem of generalized Markov numbers that are prime powers, arXiv:2312.07329 [math.NT], 2023. See Appendix A.
Kristin DeVleming and Nikita Singh, Rational unicuspidal plane curves of low degree, arXiv:2311.15922 [math.AG], 2023. See p. 14.
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
KEYWORD
nonn
AUTHOR
Paul Muljadi, Jan 01 2011
STATUS
approved