|
| |
|
|
A002559
|
|
Markoff (or Markov) numbers: union of numbers x, y, z satisfying x^2 + y^2 + z^2 = 3xyz.
(Formerly M1432 N0566)
|
|
15
|
|
|
|
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)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
A004280 gives indices of Fibonacci numbers (A000045) which are also Markoff (or Markov) numbers.
As mentioned by Conway and Guy, all odd-index 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) Markoff numbers below x, for C = 0.18071704711507....
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) says "it will be shown that the largest member of [a Markoff] triple determines the other two uniquely. This answers a question which has been open for 100 years." - Jonathan Sondow, Aug 21 2012
|
|
|
REFERENCES
|
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 187.
R. Descombes, Problemes d'approximation diophantienne. Enseignement Math. (2) 6 1960 18-26.
R. K. Guy, Don't try to solve these problems, Amer. Math. Monthly, 90 (1983), 35-41.
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)
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.
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).
Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709-723.
Ying Zhang, Congruence and uniqueness of certain Markov numbers, Acta Arithmetica 128 (2007), 295-301.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=1..1000
T. Ace, Markov numbers
M. L. Lang & S. P. Tan, A simple proof of the Markoff conjecture for prime powers
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 2012.
M. Waldschmidt, Open Diophantine problems
Y. Zhang, An Elementary Proof of Markoff Conjecture for Prime Powers
|
|
|
MATHEMATICA
|
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 *)
|
|
|
CROSSREFS
|
Sequence in context: A216378 A193044 A122491 * A049097 A045366 A158708
Adjacent sequences: A002556 A002557 A002558 * A002560 A002561 A002562
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
N. J. A. Sloane and J. H. Conway (conway(AT)math.princeton.edu)
|
|
|
EXTENSIONS
|
Zagier comment and reference from Charles R Greathouse IV, Mar 14 2010
|
|
|
STATUS
|
approved
|
| |
|
|
