|
%I M1432 N0566 #214 Jan 08 2024 09:03:07
%S 1,2,5,13,29,34,89,169,194,233,433,610,985,1325,1597,2897,4181,5741,
%T 6466,7561,9077,10946,14701,28657,33461,37666,43261,51641,62210,75025,
%U 96557,135137,195025,196418,294685,426389,499393,514229,646018,925765,1136689,1278818
%N Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3*x*y*z.
%C A004280 gives indices of Fibonacci numbers (A000045) which are also Markoff (or Markov) numbers.
%C 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 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 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 The odd numbers in this sequence are of the form 4k+1. - _Paul Muljadi_, Jan 31 2011
%C All prime divisors of Markov numbers (with exception 2) are of the form 4k+1. - _Artur Jasinski_, Nov 20 2011
%C 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 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 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 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 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 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 Named after the Russian mathematician Andrey Andreyevich Markov (1856-1922). - _Amiram Eldar_, Jun 10 2021
%D 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 John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 187.
%D Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 86.
%D Richard K. Guy, Unsolved Problems in Number Theory, D12.
%D 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 Florian Luca and A. Srinivasan, Markov equation with Fibonacci components, Fib. Q., 56 (No. 2, 2018), 126-129.
%D Richard A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002559/b002559.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 Tom Ace, <a href="https://minortriad.com/markoff.html">Markoff numbers</a>.
%H 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Norbert Riedel, <a href="http://arxiv.org/abs/1208.4032">On the Markoff Equation</a>, arXiv:1208.4032 [math.NT], 2012-2015.
%H 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 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 Anitha Srinivasan, <a href="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 Michel Waldschmidt, <a href="https://arxiv.org/abs/math/0312440">Open Diophantine problems</a>, arXiv:math/0312440 [math.NT], 2003-2004.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MarkovNumber.html">Markov Number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Markov_number">Markov number</a>.
%H 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 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 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 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 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 MAX=10^10;
%t 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 Take[data//Flatten//Union, 50] (* _Xianwen Wang_, Aug 22 2021 *)
%Y Cf. A178444, A256395.
%Y Cf. A000045, A001653, A004280, A158381, A158384,
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_ and _J. H. Conway_
%E Name clarified by _Wolfdieter Lang_, Jan 22 2015
|
