Markov number
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation
studied by Andrey Markoff (1879, 1880).
The first few Markov numbers are
appearing as coordinates of the Markov triples
- (1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), (1, 13, 34), (1, 34, 89), (2, 29, 169), (5, 13, 194), (1, 89, 233), (5, 29, 433), (1, 233, 610), (2, 169, 985), (13, 34, 1325), etc.
There are infinitely many Markov numbers and Markov triples.
Markov tree
There are two simple ways to obtain a new Markov triple from an old one (x, y, z). First, one may permute the 3 numbers x,y,z, so in particular one can normalize the triples so that x ≤ y ≤ z. Second, if (x, y, z) is a Markov triple then by Vieta jumping so is (x, y, 3xy − z). Applying this operation twice returns the same triple one started with. Joining each normalized Markov triple to the 1, 2, or 3 normalized triples one can obtain from this gives a graph starting from (1,1,1) as in the diagram. This graph is connected; in other words every Markov triple can be connected to (1,1,1) by a sequence of these operations.[1] If we start, as an example, with (1, 5, 13) we get its three neighbors (5, 13, 194), (1, 13, 34) and (1, 2, 5) in the Markov tree if z is set to 1, 5 and 13, respectively. For instance, starting with (1, 1, 2) and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers. Starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers.
All the Markov numbers on the regions adjacent to 2's region are odd-indexed Pell numbers (or numbers n such that 2n2 − 1 is a square,
where Fx is the xth Fibonacci number. Likewise, there are infinitely many Markov triples of the form
where Px is the xth Pell number.[2]
Other properties
Aside from the two smallest singular triples (1,1,1) and (1,1,2), every Markov triple consists of three distinct integers.[3]
The unicity conjecture states that for a given Markov number c, there is exactly one normalized solution having c as its largest element: proofs of this conjecture have been claimed but none seems to be correct.[4]
Odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32.[5]
In his 1982 paper, Don Zagier conjectured that the nth Markov number is asymptotically given by
Moreover, he pointed out that , an approximation of the original Diophantine equation, is equivalent to with f(t) = arcosh(3t/2).[6] The conjecture was proved by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry.[7]
The nth Lagrange number can be calculated from the nth Markov number with the formula
The Markov numbers are sums of (non-unique) pairs of squares.
Markov's theorem
Markoff (1879, 1880) showed that if
is an indefinite binary quadratic form with real coefficients and discriminant , then there are integers x, y for which f takes a nonzero value of absolute value at most
unless f is a Markov form:[8] a constant times a form
where (p, q, r) is a Markov triple and
There is also a Markov theorem in topology, named after the son of Andrey Markov, Andrei Andreevich Markov.[9]
Matrices
so that if Tr(X⋅Y⋅X−1 ⋅ Y−1) = −2 then
- Tr(X) Tr(Y) Tr(X⋅Y) = Tr(X)2 + Tr(Y)2 + Tr(X⋅Y)2
In particular if X and Y also have integer entries then Tr(X)/3, Tr(Y)/3, and Tr(X⋅Y)/3 are a Markov triple. If X⋅Y⋅Z = 1 then Tr(X⋅Y) = Tr(Z), so more symmetrically if X, Y, and Z are in SL2(Z) with X⋅Y⋅Z = 1 and the commutator of two of them has trace −2, then their traces/3 are a Markov triple.[10]
See also
Notes
- ↑ Cassels (1957) p.28
- ↑
A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5. - ↑ Cassels (1957) p.27
- ↑ Guy (2004) p.263
- ↑ Zhang, Ying (2007). "Congruence and Uniqueness of Certain Markov Numbers". Acta Arithmetica. 128 (3): 295–301. arXiv:math/0612620. Bibcode:2007AcAri.128..295Z. doi:10.4064/aa128-3-7. MR 2313995.
- ↑ Zagier, Don B. (1982). "On the Number of Markoff Numbers Below a Given Bound". Mathematics of Computation. 160 (160): 709–723. doi:10.2307/2007348. JSTOR 2007348. MR 0669663.
- ↑ Greg McShane; Igor Rivin (1995). "Simple curves on hyperbolic tori". Comptes Rendus de l'Académie des Sciences, Série I. 320 (12).
- ↑ Cassels (1957) p.39
- ↑ Louis H. Kauffman, Knots and Physics, p. 95, ISBN 978-9814383011
- ↑ Aigner, Martin (2013), "The Cohn tree", Markov's theorem and 100 years of the uniqueness conjecture, Springer, pp. 63–77, doi:10.1007/978-3-319-00888-2_4, ISBN 978-3-319-00887-5, MR 3098784 .
References
- Cassels, J.W.S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. 45. Cambridge University Press. Zbl 0077.04801.
- Cusick, Thomas; Flahive, Mari (1989). The Markoff and Lagrange spectra. Math. Surveys and Monographs. 30. Providence, RI: American Mathematical Society. ISBN 0-8218-1531-8. Zbl 0685.10023.
- Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 263–265. ISBN 0-387-20860-7. Zbl 1058.11001.
- Malyshev, A.V. (2001) [1994], "Markov spectrum problem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Markoff, A. "Sur les formes quadratiques binaires indéfinies". Mathematische Annalen. Springer Berlin / Heidelberg. ISSN 0025-5831.
- Markoff, A. (1879). "First memory". Mathematische Annalen. 15 (3–4): 381–406. doi:10.1007/BF02086269.
- Markoff, A. (1880). "Second memory". Mathematische Annalen. 17 (3): 379–399. doi:10.1007/BF01446234.