A178444 -id:A178444 - 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

A256395 Composite Markoff numbers.

Original entry on oeis.org

34, 169, 194, 610, 985, 1325, 4181, 6466, 9077, 10946, 14701, 37666, 51641, 62210, 75025, 135137, 195025, 196418, 294685, 499393, 646018, 925765, 1136689, 1278818, 1346269, 1441889, 2012674, 2423525, 3524578, 4400489, 6625109, 7453378, 8399329, 9227465, 9647009

Offset: 1

Jonathan Sondow, Apr 30 2015

nonn

Bourgain, Gamburd, and Sarnak have announced a proof that almost all Markoff numbers A002559 are composite. Equivalently, the prime Markoff numbers A178444 have density zero among all Markoff numbers. (It is conjectured that infinitely many Markoff numbers are prime.)

See A002559 for references, links, and additional comments.

J. Bourgain, A. Gamburd, and P. Sarnak, Markoff triples and strong approximation, arXiv:1505.06411 [math.NT], 2015.

Intersection of A002808 and A002559.

Complement of (A178444 union {1}) in A002559.

Rest[Select[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, 50], ! PrimeQ[#] &]]

def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:
    # Using function 'MarkovNumbers' from A002559.
    M = MarkovNumbers(len, MAX)
    U = set([])
    for m in M:
        if not is_prime(ZZ(m)):
            U.add(m)
    return sorted(U)[1:len]
# Balance required sequence length and search depth.
print(A386894List(len=56))  # Peter Luschny, Aug 12 2025

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

A173957 Numbers n such that x^2+y^2+prime(n)^2=3xy*prime(n).

Original entry on oeis.org

1, 3, 6, 10, 24, 51, 84, 251, 419, 755, 960, 3121, 3582, 4515, 9299, 35878, 42613, 127165, 211625, 235352, 5457937, 17353542, 23023556

Offset: 1

Juri-Stepan Gerasimov, Mar 03 2010

nonn

			a(1)=1 because 1^2+5^2+prime(1)^2=3*1*5*prime(1); a(2)=3 because 1^2+2^2+prime(3)^2=3*1*2*prime(3); a(3)=6 because 29^2+34^2+prime(6)^2=3*29*34*prime(6); a(4)=10 because 13^2+34^2=prime(10)^2=3*13*34*prime(10).

Cf. A000040, A002559, A178444 (prime(n)).

{n: A000040(n) in A002559}. [From R. J. Mathar, Mar 09 2010]

Extended beyond 960 by R. J. Mathar, Mar 09 2010

A182585 Markov numbers that are semiprime.

Original entry on oeis.org

34, 169, 194, 985, 4181, 9077, 14701, 51641, 135137, 294685, 499393, 646018, 925765, 1136689, 1346269, 2012674, 6625109, 8399329, 9647009, 11485154, 16964653, 21531778, 43484701, 48928105, 111242465, 137295677, 144059117, 165580141, 225058681, 253191266

Offset: 1

Jonathan Vos Post, May 06 2012

This is to A178444 as semiprimes A001358 are to primes A000040.

			194 is in this sequence because it is a Markoff (or Markov) number, and 194 = 2 * 97.

Cf. A001358, A002559, A178444.

A001358 Intersection A002559. {x, y, or z satisfying x^2 + y^2 + z^2 = 3xyz, also such that A001222(of that number) = 2}.

A339062 Sorted list of prime numbers in the union of 7-tuples (a,b,c,d,e,f,g) satisfying a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = abcdefg.

Original entry on oeis.org

2, 3, 5, 23, 37, 67, 181, 307, 359, 1559, 2417, 59123, 88327, 95783, 99907, 304151, 606839, 932999, 1179491, 1531619, 1860337, 2188919, 2363441, 3578437, 5474849, 7577351, 11273459, 12994823, 32393057, 48222721, 127896599, 248648401, 932998067, 1109123111, 2671715093, 4488932999, 9347244311

Offset: 1

Giorgos Kalogeropoulos, Nov 22 2020

nonn

Prime numbers that appear in the integer solutions {X(1),X(2),...X(n)} of Markoff-Hurwitz equation X(1)^2 + ... + X(n)^2 = a*X(1)*...*X(n) for a=1 and n=7.
7-tuples that are solutions of the above equation consisting only of primes appear to be very rare. In this special case the number N=X(1)*...*X(7) is equal to the sum of the squares of its prime factors (with multiplicity).
Giorgos Kalogeropoulos has found two numbers N having 123 and 163 digits respectively.
The factors of the first one are {2, 2, 2, 23, 1109123111, 57766182616657495290612267717977498812931942308391, 11788844704086155814066994795339207139099517865226893357415731}, so this 7-tuple is a solution and all these primes belong to the sequence. (See Rivera's link for the second 7-tuple).

			{1, 1, 2, 2, 3, 23, 274} is a solution to the equation. So the primes 2,3,23 are terms of the sequence.

Carlos Rivera, Puzzle 1019. Follow-up to Puzzle 625, The Prime Puzzles and Problems Connection.

Cf. A227208, A178444.

cp's OEIS Frontend

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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A256395 Composite Markoff numbers.

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

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A173957 Numbers n such that x^2+y^2+prime(n)^2=3*x*y*prime(n).

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A182585 Markov numbers that are semiprime.

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A339062 Sorted list of prime numbers in the union of 7-tuples (a,b,c,d,e,f,g) satisfying a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = a*b*c*d*e*f*g.

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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.

A173957 Numbers n such that x^2+y^2+prime(n)^2=3xy*prime(n).

A339062 Sorted list of prime numbers in the union of 7-tuples (a,b,c,d,e,f,g) satisfying a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = abcdefg.