By Martin Aigner

This e-book takes the reader on a mathematical trip, from a number-theoretic perspective, to the world of Markov’s theorem and the individuality conjecture, progressively unfolding many appealing connections till every little thing falls into position within the evidence of Markov’s theorem. What makes the Markov subject so beautiful is that it seems that in an magnificent number of assorted fields, from quantity conception to combinatorics, from classical teams and geometry to the area of graphs and words.

On the best way, there also are introductory forays into a few attention-grabbing issues that don't belong to the traditional curriculum, corresponding to Farey fractions, modular and loose teams, hyperbolic planes, and algebraic phrases. The ebook closes with a dialogue of the present kingdom of data concerning the forte conjecture, which is still an open problem to this day.

All the fabric will be obtainable to upper-level undergraduates with a few historical past in quantity concept, and whatever past this point is absolutely defined within the text.

This isn't a monograph within the traditional feel targeting a particular subject. as a substitute, it narrates in 5 elements – Numbers, timber, teams, phrases, Finale – the tale of a discovery in a single box and its many manifestations in others, as a tribute to an exceptional mathematical success and as an highbrow excitement, considering the marvellous team spirit of all mathematics.

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Additional info for Markov's Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings

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26 will finish the proof. 3 The Uniqueness Conjecture √ 9+ 221 10 and the triple (29, 2, 5) with m = 29, u = 12 yields √ 7565 29 . A Markov form fm (x, y) and also the numbers γm were defined in terms of a Markov triple (m, m2 , m3 ), where m is the maximum of the three numbers involved. Stated in this way, there is an obvious ambiguity about the definition, leading to two questions: 1. Does every Markov number m appear as the maximum in a Markov triple? 2. Could it be that some m ∈ M appears as the maximum in more than one Markov triple, so that fm resp.

He died in Berlin in 1917. 40 Markov’s Theorem and the Uniqueness Conjecture Notes The classical papers on Markov’s theorem and the uniqueness conjecture are Markov [69, 70] in 1879/80 and Frobenius [43] in 1913. Even earlier (1873), Korkine and Zolotareff [59] proved that the first two Lagrange numbers are √ √ 5 and 8, and Markov acknowledged their work as inspiration for his own. Markov used complicated calculations of continued fractions in his proof. Frobenius studied in detail the Markov forms fm (x, y), but could not show that every form f with M(f ) < 3 is equivalent to some fm .

8. Suppose c d a b between and denominator. a b a b < a b are Farey neighbors. Then among all fractions , the mediant a+a b+b is the unique fraction with smallest Proof. 9), b+b a b − ab 1 ≤ = , dbb bb bb which implies d ≥ b + b . 10) becomes an equality, and we get a d − b c = 1, cb − da = 1. Solving for c and d, we obtain c = a + a , d = b + b , which proves the uniqueness part. Now we come to the most important property of the Farey table. 9. Every rational number t between 0 and 1 appears in the Farey table, and every t = 0, 1 is generated as a mediant exactly once.

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