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Markov's Theorem and 100 Years of the Uniqueness Conjecture

Martin Aigner
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
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The title of this book very precisely states its subject, so the reader might expect to find the statement of Markov’s theorem on the first page. However, the theorem is not simple to state, and therefore it is not stated before Chapter 2. The necessary preparation is not without its own pleasures. For instance, this reviewer loved the fact that the number \[\sum_{n\geq 1} \frac{1}{10^{n!}} \] is transcendental, and its very simple proof.

In order to state Markov’s theorem, consider a real number \(\alpha\), and let us try to approximate it by rationals \(p/q\). Consider all real numbers \(L>0\) such that \[ \left |\,\alpha -\frac{p}{q} \right | < \frac{1}{Lq^2} \] for infinitely many rationals \(p/q\). The supremum of all such real numbers \(L\) is denoted by \(L(\alpha)\), and is called the Lagrange number of \(\alpha\). Finally, the set \(\{L(\alpha) : \alpha \in \mathbf{R} \setminus \mathbf{Q} \}\) is called the Lagrange spectrum.

It turns out that the interesting part of the Lagrange spectrum is its subset \(\mathcal{L}_3\) consisting the \(L(\alpha)\) that are less than 3. Now consider Markov’s equation, that is, the equation \[ x_1^2 + x_2^2 +x_3^2 = 3x_1x_2x_3.\] Positive integer solutions to this equation are called Markov triples: examples are \((1,1,1)\), \((1, 2, 5)\), and \((1, 5, 13)\). Integers that are part of a Markov triple are called Markov numbers. The first few Markov numbers are 1, 2, 5, 13, 29, and 34. The set of Markov numbers is denoted by \(M\).

Markov’s theorem states that \[{\cal L}_3 = \left \{\frac{\sqrt{9m^2 - 4}}{m} : m\in M \right \} .\]

The Uniqueness Conjecture claims that every Markov number occurs exactly once as the maximum of a Markov triple. Interestingly, this conjecture has an equivalent version that does not even mention Markov numbers. There have been numerous authors who claimed to have proved the conjecture but whose arguments turned out to be flawed.

In the rest of the book, the author surveys the diverse parts of mathematics that contribute to the proof of Markov’s theorem, such as group theory, combinatorics, and linear algebra. He considers the interplay of these fields in proving Markov’s theorem a glorious moment that shows the unity of mathematics. Finally, he proves Markov’s theorem, and explains the current state of affairs for the Uniqueness Conjecture.

The book benefits greatly from the authors ability to choose great examples. It is unlikely that classes will be taught from the book since it is too specialized, but a chapter here or there can be assigned for a reading project or seminar presentation.

Miklós Bóna is Professor of Mathematics at the University of Florida.

Approximation of Irrational Numbers

Markov's Theorem and the Uniqueness Conjecture

The Markov Tree

The Cohn Tree

The Modular Group SL(2,Z)

The Free Group F2

Christoffel Words

Sturmian Words

Proof of Markov’s Theorem

The Uniqueness Conjecture. ​