By William Cook; et al
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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  in 1913. Even earlier (1873), Korkine and Zolotareﬀ  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.