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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
8
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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.
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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)
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LINKS
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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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Adjacent sequences: A002556 A002557 A002558 this_sequence A002560 A002561 A002562
Sequence in context: A126656 A026522 A122491 this_sequence A049097 A045366 A093702
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KEYWORD
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nonn,nice,easy
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AUTHOR
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njas and J. H. Conway (conway(AT)math.princeton.edu)
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