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Search: id:A002559
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| A002559 |
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Markoff (or Markov) numbers: union of numbers x, y, z satisfying x^2 + y^2 + z^2 = 3xyz.
(Formerly M1432 N0566)
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+0
10
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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
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REFERENCES
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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)
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).
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LINKS
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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
M. Waldschmidt, Open Diophantine problems
Y. Zhang, An Elementary Proof of Markoff Conjecture for Prime Powers
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MATHEMATICA
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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] (from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 05 2005)
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CROSSREFS
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Sequence in context: A126656 A026522 A122491 this_sequence A049097 A045366 A158708
Adjacent sequences: A002556 A002557 A002558 this_sequence A002560 A002561 A002562
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
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