A178444 Markov numbers that are prime.

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

Offset: 1

Author

Paul Muljadi, Jan 01 2011

Keywords

• nonn

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.

Crossrefs

Cf. A002559, A256395.

Programs

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]