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

%I A002559 M1432 N0566 #234 Aug 12 2025 14:55:41
%S A002559 1,2,5,13,29,34,89,169,194,233,433,610,985,1325,1597,2897,4181,5741,
%T A002559 6466,7561,9077,10946,14701,28657,33461,37666,43261,51641,62210,75025,
%U A002559 96557,135137,195025,196418,294685,426389,499393,514229,646018,925765,1136689,1278818
%N A002559 Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3*x*y*z.
%C A002559 A004280 gives indices of Fibonacci numbers (A000045) which are also Markoff (or Markov) numbers.
%C A002559 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
%C A002559 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
%C A002559 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]
%C A002559 The odd numbers in this sequence are of the form 4k+1. - _Paul Muljadi_, Jan 31 2011
%C A002559 All prime divisors of Markov numbers (with exception 2) are of the form 4k+1. - _Artur Jasinski_, Nov 20 2011
%C A002559 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
%C A002559 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
%C A002559 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
%C A002559 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
%C A002559 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
%C A002559 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
%C A002559 Named after the Russian mathematician Andrey Andreyevich Markov (1856-1922). - _Amiram Eldar_, Jun 10 2021
%D A002559 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
%D A002559 John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 187.
%D A002559 Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 86.
%D A002559 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.3, p. 200.
%D A002559 Richard K. Guy, Unsolved Problems in Number Theory, D12.
%D A002559 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)
%D A002559 Florian Luca and A. Srinivasan, Markov equation with Fibonacci components, Fib. Q., 56 (No. 2, 2018), 126-129.
%D A002559 Richard A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125.
%D A002559 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002559 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002559 T. D. Noe, <a href="/A002559/b002559.txt">Table of n, a(n) for n = 1..1000</a>
%H A002559 Ryuji Abe and Benoît Rittaud, <a href="https://doi.org/10.1016/j.disc.2017.07.010">On palindromes with three or four letters associated to the Markoff spectrum</a>, Discrete Mathematics, Vol. 340, No. 12 (2017), pp. 3032-3043.
%H A002559 Tom Ace, <a href="https://minortriad.com/markoff.html">Markoff numbers</a>.
%H A002559 Enrico Bombieri, <a href="http://dx.doi.org/10.1016/j.exmath.2006.10.002">Continued fractions and the Markoff tree</a>, Expo. Math., Vol. 25, No. 3 (2007), pp. 187-213.
%H A002559 Jean Bourgain, Alex Gamburd, and Peter Sarnak, <a href="https://doi.org/10.1016/j.crma.2015.12.006">Markoff triples and strong approximation</a>, Comptes Rendus Mathematique, Vol. 354, No. 2 (2016), pp. 131-135; <a href="http://arxiv.org/abs/1505.06411">arXiv preprint</a>, arXiv:1505.06411 [math.NT], 2015.
%H A002559 Roger Descombes, <a href="http://dx.doi.org/10.5169/seals-36333">Problèmes d'approximation diophantienne</a>, L'Enseignement Math. (2), Vol. 6 (1960), pp. 18-26.
%H A002559 Roger Descombes, <a href="/A002559/a002559.pdf">Problèmes d'approximation diophantienne</a>, L'Enseignement Math. (2), Vol. 6 (1960), pp. 18-26. [Annotated scanned copy]
%H A002559 Jonathan David Evans and Ivan Smith, <a href="https://doi.org/10.2140/gt.2018.22.1143">Markov numbers and Lagrangian cell complexes in the complex projective plane</a>, Geometry & Topology, Vol. 22 (2018), pp. 1143-1180; <a href="https://arxiv.org/abs/1606.08656">arXiv preprint</a>, arXiv:1606.08656 [math.SG], 2016-2017.
%H A002559 Sam Evans, Perrine Jouteur, Sophie Morier-Genoud, and Valentin Ovsienko, <a href="https://hal.science/hal-05186307">On q-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges</a>, hal-05186307 (2025). See p. 2.
%H A002559 Georg Frobenius, <a href="https://edoc.bbaw.de/frontdoor/index/index/docId/5095">Über die Markoffschen Zahlen</a>, Sitzungsberichte der Königlichen Preußischen Akademie der Wissenschaften, Jahrgang 1913.
%H A002559 Carlos A. Gómez, Jhonny C. Gómez, and Florian Luca, <a href="https://doi.org/10.33039/ami.2020.06.001">Markov triples with k-generalized Fibonacci components</a>, Annales Mathematicae et Informaticae, Vol. 52 (2020), pp. 107-115.
%H A002559 Richard K. Guy, <a href="http://www.jstor.org/stable/2975688">Don't try to solve these problems</a>, Amer. Math. Monthly, Vol. 90, No. 1 (1983), pp. 35-41.
%H A002559 Yasuaki Gyoda, <a href="https://arxiv.org/abs/2109.09639">Positive integer solutions to (x+y)^2+(y+z)^2+(z+x)^2=12xyz</a>, arXiv:2109.09639 [math.NT], 2021.
%H A002559 Hayder Raheem Hashim and Szabolcs Tengely, <a href="https://doi.org/10.1515/ms-2017-0414">Solutions of a generalized markoff equation in Fibonacci numbers</a>, Mathematica Slovaca, Vol. 70, No. 5 (2020), pp. 1069-1078.
%H A002559 Masanobu Kaneko, <a href="http://www.ams.org/amsmtgs/2190_abstracts/1078-11-124.pdf">Congruences of Markoff numbers via Farey parametrization</a>, Preliminary Report, Dec 2011, AMS 1078-11-124, listed in Abstracts of Papers Presented to AMS, Vol.33, No.2, Issue 168, Spring 2012.
%H A002559 Sebastien Labbé, Mélodie Lapointe, and Wolfgang Steiner, <a href="https://arxiv.org/abs/2212.09852">A q-analog of the Markoff injectivity conjecture holds</a>, arXiv:2212.09852 [math.CO], 2022.
%H A002559 Clément Lagisquet, Edita Pelantová, Sébastien Tavenas, and Laurent Vuillon, <a href="https://arxiv.org/abs/2010.10335">On the Markov numbers: fixed numerator, denominator, and sum conjectures</a>, arXiv:2010.10335 [math.CO], 2020.
%H A002559 Mong Lung Lang and Ser Peow Tan, <a href="https://doi.org/10.1007/s10711-007-9189-x">A simple proof of the Markoff conjecture for prime powers</a>, Geometriae Dedicata, Vol. 129 (2007), pp. 15-22; <a href="https://arxiv.org/abs/math/0508443">arXiv preprint</a>, arXiv:math/0508443 [math.NT], 2005.
%H A002559 Kyungyong Lee, Li Li, Michelle Rabideau, and Ralf Schiffler, <a href="https://arxiv.org/abs/2010.13010">On the ordering of the Markov numbers</a>, arXiv:2010.13010 [math.NT], 2020.
%H A002559 James Propp, <a href="http://faculty.uml.edu/jpropp/markoff-talk.html">The combinatorics of Markov numbers</a>, U. Wisconsin Combinatorics Seminar, April 4, 2005.
%H A002559 S. G. Rayaguru, M. K. Sahukar, and G. K. Panda, <a href="https://doi.org/10.7546/nntdm.2020.26.3.149-159">Markov equation with components of some binary recurrent sequences</a>, Notes on Number Theory and Discrete Mathematics, Vol. 26, No. 3 (2020), pp. 149-159.
%H A002559 Norbert Riedel, <a href="http://arxiv.org/abs/1208.4032">On the Markoff Equation</a>, arXiv:1208.4032 [math.NT], 2012-2015.
%H A002559 Julieth F. Ruiz, Jose L. Herrera, and Jhon J. Bravo, <a href="https://doi.org/10.3390/math12010108">Markov Triples with Generalized Pell Numbers</a>, Mathematics 12, 108, (2024).
%H A002559 Anitha Srinivasan, <a href="http://dx.doi.org/10.5802/jtnb.701">Markoff numbers and ambiguous classes</a>, Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), pp. 757-770.
%H A002559 Anitha Srinivasan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/58-5/srinivasan.pdf">The Markoff-Fibonacci Numbers</a>, Fibonacci Quart., Vol. 58, No. 5 (2020), pp. 222-228.
%H A002559 Michel Waldschmidt, <a href="https://arxiv.org/abs/math/0312440">Open Diophantine problems</a>, arXiv:math/0312440 [math.NT], 2003-2004.
%H A002559 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MarkovNumber.html">Markov Number</a>.
%H A002559 Wikipedia, <a href="https://en.wikipedia.org/wiki/Markov_number">Markov number</a>.
%H A002559 Don Zagier, <a href="http://dx.doi.org/10.1090/S0025-5718-1982-0669663-7">On the number of Markoff numbers below a given bound</a>, Mathematics of Computation, Vol. 39, No. 160 (1982), pp. 709-723.
%H A002559 Ying Zhang, <a href="https://arxiv.org/abs/math/0606283">An Elementary Proof of Markoff Conjecture for Prime Powers</a>, arXiv:math/0606283 [math.NT], 2006-2007.
%H A002559 Ying Zhang, <a href="http://dx.doi.org/10.4064/aa128-3-7">Congruence and uniqueness of certain Markov numbers</a>, Acta Arithmetica, Vol. 128 (2007), pp. 295-301.
%t A002559 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 *)
%t A002559 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 *)
%t A002559 MAX=10^10;
%t A002559 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[#]<MAX&]&, {{1, 1, 1}, {1, 1, 2}}, UnsameQ, 2];
%t A002559 Take[data//Flatten//Union, 50] (* _Xianwen Wang_, Aug 22 2021 *)
%o A002559 (Python)
%o A002559 markov = set[tuple[int, int, int]]
%o A002559 def MarkovNumbers(len: int = 50, MAX: int = 10**10) -> list[int]:
%o A002559     cur: markov = {(1, 1, 1), (1, 1, 2), }
%o A002559     def step(triples: markov) -> markov:
%o A002559         ret: markov = set()
%o A002559         for (a, b, c) in triples:
%o A002559             for x, y, z in [(a, b, c), (b, c, a), (c, a, b)]:
%o A002559                 t = (x, y, 3 * x * y - z)
%o A002559                 if max(t) < MAX: ret.add(t)
%o A002559         return ret
%o A002559     while True:
%o A002559         new = step(cur)
%o A002559         if new == cur: break
%o A002559         cur = new
%o A002559     return sorted({n for triple in cur for n in triple})[:len]
%o A002559 print(MarkovNumbers(len=42))  # _Peter Luschny_, Aug 10 2025
%Y A002559 Cf. A178444, A256395, A000045, A001653, A004280, A158381, A158384.
%K A002559 nonn,nice,easy
%O A002559 1,2
%A A002559 _N. J. A. Sloane_ and _J. H. Conway_
%E A002559 Name clarified by _Wolfdieter Lang_, Jan 22 2015