A002559 Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3*x*y*z.

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, 1597, 2897, 4181, 5741, 6466, 7561, 9077, 10946, 14701, 28657, 33461, 37666, 43261, 51641, 62210, 75025, 96557, 135137, 195025, 196418, 294685, 426389, 499393, 514229, 646018, 925765, 1136689, 1278818

Offset: 1

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• nice
• easy

Comments

A004280 gives indices of Fibonacci numbers (A000045) which are also Markoff (or Markov) numbers.

As mentioned by Conway and Guy, all odd-indexed Pell numbers (A001653) also appear in this sequence. The positions of the Fibonacci and Pell numbers in this sequence are given in A158381 and A158384, respectively. - T. D. Noe, Mar 19 2009

Assuming that each solution (x,y,z) is ordered x <= y <= z, the open problem is to prove that each z value occurs only once. There are no counterexamples in the first 1046858 terms, which have z values < Fibonacci(5001) = 6.2763...*10^1044. - T. D. Noe, Mar 19 2009

Zagier shows that there are C log^2 (3x) + O(log x (log log x)^2) Markoff numbers below x, for C = 0.180717.... - Charles R Greathouse IV, Mar 14 2010 [but see Thompson, below]

The odd numbers in this sequence are of the form 4k+1. - Paul Muljadi, Jan 31 2011

All prime divisors of Markov numbers (with exception 2) are of the form 4k+1. - Artur Jasinski, Nov 20 2011

Kaneko extends a parameterization of Markoff numbers, citing Frobenius, and relates it to a conjectured behavior of the elliptic modular j-function at real quadratic numbers. - Jonathan Vos Post, May 06 2012

Riedel (2012) claims a proof of the unicity conjecture: "it will be shown that the largest member of [a Markoff] triple determines the other two uniquely." - Jonathan Sondow, Aug 21 2012

There are 93 terms with each term <= 2*10^9 in the sequence. The number of distinct prime divisors of any of the first 93 terms is less than 6. The second, third, 4th, 5th, 6th, 10th, 11th, 15th, 16th, 18th, 20th, 24th, 25th, 27th, 30th, 36th, 38th, 45th, 48th, 49th, 69th, 79th, 81st, 86th, 91st terms are primes. - Shanzhen Gao, Sep 18 2013

Bourgain, Gamburd, and Sarnak have announced a proof that almost all Markoff numbers are composite--see A256395. Equivalently, the prime Markoff numbers A178444 have density zero among all Markoff numbers. (It is conjectured that infinitely many Markoff numbers are prime.) - Jonathan Sondow, Apr 30 2015

According to Sarnak on Apr 30 2015, all claims to have proved the unicity conjecture have turned out to be false. - Jonathan Sondow, May 01 2015

The numeric value of C = lim (number of Markoff numbers < x) / log^2(3x) given in Zagier's paper and quoted above suffers from an accidentally omitted digit and rounding errors. The correct value is C = 0.180717104711806... (see A261613 for more digits). - Christopher E. Thompson, Aug 22 2015

Named after the Russian mathematician Andrey Andreyevich Markov (1856-1922). - Amiram Eldar, Jun 10 2021

References

• Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013. x+257 pp. ISBN: 978-3-319-00887-5; 978-3-319-00888-2 MR3098784
• John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 187.
• Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 86.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.3, p. 200.
• Richard K. Guy, Unsolved Problems in Number Theory, D12.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, notes on ch. 24.6 (p. 412)
• Florian Luca and A. Srinivasan, Markov equation with Fibonacci components, Fib. Q., 56 (No. 2, 2018), 126-129.
• Richard A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Ryuji Abe and Benoît Rittaud, On palindromes with three or four letters associated to the Markoff spectrum, Discrete Mathematics, Vol. 340, No. 12 (2017), pp. 3032-3043.
• Tom Ace, Markoff numbers.
• Enrico Bombieri, Continued fractions and the Markoff tree, Expo. Math., Vol. 25, No. 3 (2007), pp. 187-213.
• Jean Bourgain, Alex Gamburd, and Peter Sarnak, Markoff triples and strong approximation, Comptes Rendus Mathematique, Vol. 354, No. 2 (2016), pp. 131-135; arXiv preprint, arXiv:1505.06411 [math.NT], 2015.
• Roger Descombes, Problèmes d'approximation diophantienne, L'Enseignement Math. (2), Vol. 6 (1960), pp. 18-26.
• Roger Descombes, Problèmes d'approximation diophantienne, L'Enseignement Math. (2), Vol. 6 (1960), pp. 18-26. [Annotated scanned copy]
• Jonathan David Evans and Ivan Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geometry & Topology, Vol. 22 (2018), pp. 1143-1180; arXiv preprint, arXiv:1606.08656 [math.SG], 2016-2017.
• Sam Evans, Perrine Jouteur, Sophie Morier-Genoud, and Valentin Ovsienko, On q-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges, hal-05186307 (2025). See p. 2.
• Georg Frobenius, Über die Markoffschen Zahlen, Sitzungsberichte der Königlichen Preußischen Akademie der Wissenschaften, Jahrgang 1913.
• Carlos A. Gómez, Jhonny C. Gómez, and Florian Luca, Markov triples with k-generalized Fibonacci components, Annales Mathematicae et Informaticae, Vol. 52 (2020), pp. 107-115.
• Richard K. Guy, Don't try to solve these problems, Amer. Math. Monthly, Vol. 90, No. 1 (1983), pp. 35-41.
• Yasuaki Gyoda, Positive integer solutions to (x+y)^2+(y+z)^2+(z+x)^2=12xyz, arXiv:2109.09639 [math.NT], 2021.
• Hayder Raheem Hashim and Szabolcs Tengely, Solutions of a generalized markoff equation in Fibonacci numbers, Mathematica Slovaca, Vol. 70, No. 5 (2020), pp. 1069-1078.
• Masanobu Kaneko, Congruences of Markoff numbers via Farey parametrization, Preliminary Report, Dec 2011, AMS 1078-11-124, listed in Abstracts of Papers Presented to AMS, Vol.33, No.2, Issue 168, Spring 2012.
• Sebastien Labbé, Mélodie Lapointe, and Wolfgang Steiner, A q-analog of the Markoff injectivity conjecture holds, arXiv:2212.09852 [math.CO], 2022.
• Clément Lagisquet, Edita Pelantová, Sébastien Tavenas, and Laurent Vuillon, On the Markov numbers: fixed numerator, denominator, and sum conjectures, arXiv:2010.10335 [math.CO], 2020.
• Mong Lung Lang and Ser Peow Tan, A simple proof of the Markoff conjecture for prime powers, Geometriae Dedicata, Vol. 129 (2007), pp. 15-22; arXiv preprint, arXiv:math/0508443 [math.NT], 2005.
• Kyungyong Lee, Li Li, Michelle Rabideau, and Ralf Schiffler, On the ordering of the Markov numbers, arXiv:2010.13010 [math.NT], 2020.
• James Propp, The combinatorics of Markov numbers, U. Wisconsin Combinatorics Seminar, April 4, 2005.
• S. G. Rayaguru, M. K. Sahukar, and G. K. Panda, Markov equation with components of some binary recurrent sequences, Notes on Number Theory and Discrete Mathematics, Vol. 26, No. 3 (2020), pp. 149-159.
• Norbert Riedel, On the Markoff Equation, arXiv:1208.4032 [math.NT], 2012-2015.
• Julieth F. Ruiz, Jose L. Herrera, and Jhon J. Bravo, Markov Triples with Generalized Pell Numbers, Mathematics 12, 108, (2024).
• Anitha Srinivasan, Markoff numbers and ambiguous classes, Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), pp. 757-770.
• Anitha Srinivasan, The Markoff-Fibonacci Numbers, Fibonacci Quart., Vol. 58, No. 5 (2020), pp. 222-228.
• Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
• Eric Weisstein's World of Mathematics, Markov Number.
• Wikipedia, Markov number.
• Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation, Vol. 39, No. 160 (1982), pp. 709-723.
• Ying Zhang, An Elementary Proof of Markoff Conjecture for Prime Powers, arXiv:math/0606283 [math.NT], 2006-2007.
• Ying Zhang, Congruence and uniqueness of certain Markov numbers, Acta Arithmetica, Vol. 128 (2007), pp. 295-301.

Crossrefs

Cf. A178444, A256395, A000045, A001653, A004280, A158381, A158384.

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 *)
terms = 40; depth0 = 10; Clear[ft]; ft[n_] := ft[n] = Module[{f}, f[] = {1, 2, 5}; f[ud___, u(*up*)] := f[ud, u] = Module[{g = f[ud]}, {g[[1]], g[[3]], 3*g[[1]]*g[[3]] - g[[2]]}]; f[ud___, d(*down*)] := f[ud, d] = Module[{g = f[ud]}, {g[[2]], g[[3]], 3*g[[2]]*g[[3]] - g[[1]]}]; f @@@ Tuples[{u, d}, n] // Flatten // Union // PadRight[#, terms]&]; ft[n = depth0]; ft[n++]; While[ft[n] != ft[n - 1], n++]; Print["depth = n = ", n]; A002559 = ft[n] (* Jean-François Alcover, Aug 29 2017 *)
MAX=10^10;
data=NestWhile[Select[Union[Sort/@Flatten[Table[{a, b, 3a b -c}/.MapThread[Rule, {{a, b, c}, #}]&/@Map[RotateLeft[ii, #]&, Range[3]], {ii, #}], 1]], Max[#]Xianwen Wang, Aug 22 2021 *)

Python

markov = set[tuple[int, int, int]]
def MarkovNumbers(len: int = 50, MAX: int = 10**10) -> list[int]:
    cur: markov = {(1, 1, 1), (1, 1, 2), }
    def step(triples: markov) -> markov:
        ret: markov = set()
        for (a, b, c) in triples:
            for x, y, z in [(a, b, c), (b, c, a), (c, a, b)]:
                t = (x, y, 3 * x * y - z)
                if max(t) < MAX: ret.add(t)
        return ret
    while True:
        new = step(cur)
        if new == cur: break
        cur = new
    return sorted({n for triple in cur for n in triple})[:len]
print(MarkovNumbers(len=42))  # Peter Luschny, Aug 10 2025

Extensions

Name clarified by Wolfdieter Lang, Jan 22 2015