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Markoff Numbers
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The Markoff equation is the diophantine equation x2 +y2 +z2 = 3xyz. A solution is called a Markoff triple. The main result in this chapter is a bijection between lower Christoffel words and Markoff triples. The bijection uses several ingredients: a special representation of the free monoid into SL2(N), the so-called Fricke relations, which relate the traces of two matrices in SL2, their product and their commutator (an equation reminiscent of the Markoff equation, as noted first by Harvey Cohn). Another lemma describes the socalled Markoff moves: they relate Markoff triples each to another. The chapter ends with a statement of the famous Frobenius conjecture: it asks whether the parametrization of Markoff numbers (that is, components of a Markoff triple), which is surjective by the theorem, is also injective.
Title: Markoff Numbers
Description:
The Markoff equation is the diophantine equation x2 +y2 +z2 = 3xyz.
A solution is called a Markoff triple.
The main result in this chapter is a bijection between lower Christoffel words and Markoff triples.
The bijection uses several ingredients: a special representation of the free monoid into SL2(N), the so-called Fricke relations, which relate the traces of two matrices in SL2, their product and their commutator (an equation reminiscent of the Markoff equation, as noted first by Harvey Cohn).
Another lemma describes the socalled Markoff moves: they relate Markoff triples each to another.
The chapter ends with a statement of the famous Frobenius conjecture: it asks whether the parametrization of Markoff numbers (that is, components of a Markoff triple), which is surjective by the theorem, is also injective.
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