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A002559 Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3xyz.
(Formerly M1432 N0566)
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 (list; graph; refs; listen; history; text; internal format)



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


Abe, Ryuji, and Benoît Rittaud. "On palindromes with three or four letters associated to the Markoff spectrum." Discrete Mathematics, 340.12 (2017): 3032-3043.

Aigner, Martin. 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

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 187.

R. Descombes, Problemes d'approximation diophantienne. L'Enseignement Math. (2) 6 1960 18-26.

R. 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)

R. 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

T. Ace, Markov numbers

E. Bombieri, Continued fractions and the Markoff tree, Expo. Math. 25, No. 3 (2007), 187-213.

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

R. Descombes, Problèmes d'approximation diophantienne, L'Enseignement Math. (2) 6 1960 18-26. [Annotated scanned copy]

Jonathan D. Evans, I. Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, arXiv preprint arXiv:1606.08656 [math.SG], 2016.

R. K. Guy, Don't try to solve these problems, Amer. Math. Monthly, 90 (1983), 35-41.

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.

M. L. Lang & S. P. Tan, A simple proof of the Markoff conjecture for prime powers, arXiv:math/0508443 [math.NT], 2005.

M. L. Lang & S. P. Tan, A Simple Proof Of The Markoff Conjecture For Prime Powers

J. Propp, The combinatorics of Markov numbers

N. Riedel, On the Markoff Equation, arXiv:1208.4032 [math.NT], 2012-2015.

Anitha Srinivasan, Markoff numbers and ambiguous classes, Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 757-770.

M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003.

Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709-723.

Y. 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 128 (2007), 295-301.


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 *)


Cf. A178444, A256395.

Sequence in context: A122491 A290194 A241392 * A049097 A045366 A158708

Adjacent sequences:  A002556 A002557 A002558 * A002560 A002561 A002562




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


Name clarified by Wolfdieter Lang, Jan 22 2015



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Last modified October 21 13:33 EDT 2017. Contains 293696 sequences.