The geometry of markoff numbers

1985 ◽  
Vol 7 (3) ◽  
pp. 20-29 ◽  
Author(s):  
Caroline Series
Keyword(s):  
Markoff Numbers

scholarly journals The uniqueness of the Markoff numbers

1976 ◽  
Vol 30 (134) ◽  
pp. 361-361 ◽  
Author(s):  
Gerhard Rosenberger
Keyword(s):  
Markoff Numbers

Numerology

2018 ◽  
pp. 69-74
Author(s):  
Christophe Reutenauer
Keyword(s):  
Quadratic Forms ◽  
Triple P ◽  
Mathematical Objects ◽  
Markoff Numbers

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen Markoff numbers <1000 are given, together with the associated mathematical objects considered before in the book:Markoff constants, Christoffel words, the associated matrices by the representation of Chapter 3, theMarkoff quadratic numbers whose expansion is given by the Christoffel word, the Markoff quadratic forms. Some results of Frobenius, Aigner, andClemens are given. In particular thematrix associated with a Christoffel word may be computed directly from its Markoff triple.

scholarly journals On the Frobenius conjecture for Markoff numbers

2013 ◽  
Vol 133 (7) ◽  
pp. 2363-2373 ◽  
Author(s):  
Feng-Juan Chen ◽  
Yong-Gao Chen
Keyword(s):  
Markoff Numbers

scholarly journals Congruence and uniqueness of certain Markoff numbers

2007 ◽  
Vol 128 (3) ◽  
pp. 295-301 ◽  
Author(s):  
Ying Zhang
Keyword(s):  
Markoff Numbers

From Christoffel Words to Markoff Numbers

2018 ◽  
Author(s):  
Christophe Reutenauer
Keyword(s):  
Free Group ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Special Class ◽  
Rational Numbers ◽  
Real Numbers ◽  
Inner Automorphisms ◽  
Sturmian Words ◽  
Markoff Numbers ◽  
Lyndon Words

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

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