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An elementary proof of uniqueness of Markoff numbers which are prime powers

07/2006;
Source: arXiv

ABSTRACT We present a very elementary proof of the uniqueness of Markoff numbers which are prime powers or twice prime powers, in the sense that it uses neither algebraic number theory nor hyperbolic geometry.

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    ABSTRACT: By making use of only simple facts about congruence, we first show that every even Markoff number is congruent to 2 modulo 32, and then, generalizing an earlier result of Baragar, establish the uniqueness for those Markoff numbers c where one of 3c - 2 and 3c + 2 is a prime power, 4 times a prime power, or 8 times a prime power.
    Acta Arithmetica 01/2007; 128(3). DOI:10.4064/aa128-3-7 · 0.42 Impact Factor
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    ABSTRACT: Let α be a real number. Dirichlet showed that there exist infinitely many fractions p q∈ℚ with |α-p q|≤1 q 2 . Now consider all positive real numbers L such that |α-p q|<1/Lq 2 holds for infinitely many fractions p q. The supremum of all these L is denoted by L(α); then L(α)=0 if and only if α is rational, and L(α)≥1 otherwise. The set ℒ={L(α):α∈ℝ∖ℚ} is called the Lagrange spectrum. A. Markoff [Math. Ann. 15, 381–407 (1879; JFM 11.0147.01); 17, 379–400 (1880; JFM 12.0143.02)] showed that the Lagrange spectrum below 3 consists of all numbers of the form 9m 2 -4/m, where m runs through the “Markoff numbers”. These are defined as the set of all natural numbers x i occurring as solutions of the Diophantine equation x 1 2 +x 2 2 +x 3 2 =3x 1 x 2 x 3 . It is easy to show that every Markoff number appears as the largest number in some Markoff triple (x 1 ,x 2 ,x 3 ), and the uniqueness conjecture predicts that each Markoff number is the maximum of a unique Markoff triple. The investigation of the uniqueness conjecture from different perspectives is the main goal of this book. After some introductory chapters, the reader is introduced to Cohn matrices, the modular group SL(2,ℤ), free groups, graphs and trees, and to partial results towards the uniqueness conjecture using the arithmetic of quadratic fields. The whole discussion is very elementary, and requires no preliminaries except some familiarity with basic concepts of algebra and number theory. The reviewer regrets that the author has given in to the temptation of keeping the book on a very elementary level throughout. Readers enjoying the section on hyperbolic geometry will be good advised to have a look at the very nice book “Fuchsian groups” by S. Katok [Fuchsian groups. Chicago: The University of Chicago Press (1992; Zbl 0753.30001)]. Similarly, Markoff’s original motivation for studying these questions, the theory of binary quadratic forms, is only briefly mentioned on pp. 36–38 even though quadratic forms cast their shadows almost everywhere in this book: continued fractions and the unimodular group are intimately connected with Lagrange reduction of quadratic forms, and the arithmetic of ideals in quadratic number fields also is just one way of presenting Gauss composition and the class group of forms. The readers will find a beautiful introduction to the dictionary between these languages in the recent book “Algebraic theory of quadratic numbers” by M. Trifkovič [Algebraic theory of quadratic numbers. New York, NY: Springer (2013; Zbl 1280.11002)]. This beautiful book gives readers a chance to familiarize themselves with a very simple and yet very difficult problem in number theory, and teaches them that it pays to look at a problem from many different angles. I recommend it to all students who are already hooked to number theory, and perhaps even more to those who are not.
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    ABSTRACT: A triple (a, b, c) of positive integers is called a Markoff triple iff it satisfies the Diophantine equation a2+b2+c2=abc . Recasting the Markoff tree, whose vertices are Markoff triples, in the framework of integral upper triangular 3x3 matrices, it will be shown that the largest member of such a triple determines the other two uniquely. This answers a question which has been open for 100 years. The solution of this problem will be obtained in the course of a broader investigation of the Markoff equation by means of 3x3 matrices.

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