A256395 -id:A256395 - cp's OEIS frontend

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

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

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

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).

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.

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

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 *)

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

Name clarified by Wolfdieter Lang, Jan 22 2015

A178444 Markov numbers that are prime.

Original entry on oeis.org

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

Paul Muljadi, Jan 01 2011

nonn

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

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.

Cf. A002559, A256395.

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]

A291694 Array of Markov triples (x,y,z) sorted by z, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 5, 1, 5, 13, 2, 5, 29, 1, 13, 34, 1, 34, 89, 2, 29, 169, 5, 13, 194, 1, 89, 233, 5, 29, 433, 1, 233, 610, 2, 169, 985, 13, 34, 1325, 1, 610, 1597, 5, 194, 2897, 1, 1597, 4181, 2, 985, 5741, 5, 433, 6466, 13, 194, 7561, 34, 89, 9077, 1, 4181, 10946, 29, 169, 14701

Offset: 1

Jean-François Alcover, Aug 30 2017

The positive integers x, y, z satisfy the Diophantine equation x^2 + y^2 + z^2 = 3*x*y*z, 1 <= x <= y <= z.

			The array of Markov triples begins:
  (1,  1,  1),
  (1,  1,  2),
  (1,  2,  5),
  (1,  5, 13),
  (2,  5, 29),
  (1, 13, 34),
  ...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.31.3 Markov-Hurwitz Equation, p. 200.

Eric Weisstein's World of Mathematics, Markov Number.
Wikipedia, Markov number.

Cf. A002559 (main entry for this sequence), A178444, A256395, A261613, A351372.

triples = 30; depth0 = 10 (* adjust depth according to message after first run *) ; Clear[zz, fx, fy]; fx[1] = fy[1] = fx[2] = fy[2] = fx[5] = 1;
fy[5] = 2; zz[n_] := zz[n] = Module[{f, x, y, z}, f[] = {1, 2, 5}; f[ud___, u(*up*)] := f[ud, u] = Module[{g = f[ud]}, x = g[[1]]; y = g[[3]]; z = 3*g[[1]]*g[[3]] - g[[2]]; fx[z] = x; fy[z] = y; {x, y, z}]; f[ud___, d(*down*)] := f[ud, d] = Module[{g = f[ud]}, x = g[[2]]; y = g[[3]]; z = 3*g[[2]]*g[[3]] - g[[1]]; fx[z] = x; fy[z] = y; {x, y, z}]; f @@@ Tuples[{u, d}, n] // Flatten // Union // PadRight[#, triples]&]; zz[n = depth0]; zz[n++]; While[zz[n] != zz[n - 1], n++]; Print["depth = n = ", n]; xyz = {fx[#], fy[#], #} & /@ zz[n]; Flatten[xyz]

N=5000;
for(k=1, N, for(j=1, k, for(i=1, j, if(i*j>k, break); if(i^2+j^2+k^2==3*i*j*k, print1(i, ", ", j, ", ", k, ", "))))); \\ Seiichi Manyama, Feb 16 2022

A386894 Markoff numbers that are powers of one odd prime or twice powers of one odd prime.

Original entry on oeis.org

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 646018, 1686049, 2012674, 2922509, 3276509, 11485154, 21531778, 94418953, 253191266, 321534781, 433494437, 780291637, 1405695061, 1475706146, 2971215073, 6684339842, 19577194573

Offset: 1

Wolfdieter Lang, Aug 07 2025

A subsequence of A000961 without numbers divisible by 4.
The powers of odd primes are given in A061345 (with offset 0).
These Markoff numbers (see A002559) have been proved to obey the Frobenius-Markoff uniqueness conjecture. See Aigner, Corollary 3.20, p. 59, and there the references [4] A. Baragar, [18] J. O. Button, and [119] Ying Zhang.

			26 = 2*13 is not a Markoff number, hence not in the present sequence.
610 = 2*5*61 is a Markoff number but not a prime power nor is 305 a prime power.

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013.

Mong Lung Lang and Ser Peow Tan, A simple proof of the Markoff conjecture for prime powers, arXiv:math/0508443 [math.NT], 2005.
Paul Schmutz, Systoles of arithmetic surfaces and the Markoff spectrum, Math. Ann. 305 (1996), no. 1, 191-203.
Ying Zhang, An elementary proof of uniqueness of Markoff numbers which are prime powers, arXiv:math/0606283 [math.NT], 2007.

Cf. A000040, A000961, A002559, A061345, A256395.

MAX=10^11; 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[#]James C. McMahon, Aug 12 2025 *)

def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:
    # Using function 'MarkovNumbers' from A002559.
    M = MarkovNumbers(len, MAX)
    U = set([1])
    for m in M:  # if m is a Markov number and ...
        z = ZZ(m)
        if is_prime_power(z) or (is_even(z) and is_prime_power(z//2)):
            U.add(m)
    return sorted(U)
# Balance required sequence length and search depth.
print(A386894List(len=120, MAX=10**12))  # Peter Luschny, Aug 12 2025

Markoff numbers of the form 2^j*p^k, with an odd prime p, j = 0 or 1, and k >= 0, ordered strictly increasing.

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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.

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A178444 Markov numbers that are prime.

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A291694 Array of Markov triples (x,y,z) sorted by z, read by rows.

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A386894 Markoff numbers that are powers of one odd prime or twice powers of one odd prime.

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