Continued fractions and orderings on the Markov numbers

Rabideau, Michelle; Schiffler, Ralf

doi:10.1016/j.aim.2020.107231

Cited by 9 publications

(10 citation statements)

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“…As we proved Proposition 18 for all couples (d, n) ∈ N 2 , it directly implies the fixed numerator conjecture which was already solved in [11]:…”

Section: Application To the Fixed Denominator And Numerator Conjecturessupporting

confidence: 69%

“…In particular, we can index all Markov numbers by the Farey fraction which stands at the same place in the Stern-Brocot tree (see [3,10]), this correspondence would be one to one if the unicity conjecture was true. We refer to a Markov number by its associated rational in the Farey tree namely m p q (following the notation in [11]). It would be very helpful to understand the growth of Markov numbers according to their indices.…”

Section: Introductionmentioning

confidence: 99%

“…The fixed numerator conjecture was proved in [11] using cluster algebra and snake graphs. The proof is quite technical with many interesting tools on discrete paths on the N 2 grid.…”

Section: Introductionmentioning

confidence: 99%

“…The proof is quite technical with many interesting tools on discrete paths on the N 2 grid. Nevertheless, one key concept is to use perfect matching on snake graphs (see [7,11]) and while the technique is fine this add too much concepts on the transformation of paths.…”

Section: Introductionmentioning

confidence: 99%

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On the Markov numbers: Fixed numerator, denominator, and sum conjectures

Lagisquet

Pelantová

Tavenas

et al. 2021

Advances in Applied Mathematics

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The Markov numbers are the positive integer solutions of the Diophantine equation x 2 + y 2 + z 2 = 3xyz. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov numbers by the rationals from [0, 1] which stand at the same place in the Stern-Brocot binary tree. The Frobenius' conjecture claims that each Markov number appears at most once in the tree.In particular, if the conjecture is true, the order of Markov numbers would establish a new strict order on the rationals. Aigner suggested three conjectures to better understand this order. The first one has already been solved for a few months. We prove that the other two conjectures are also true.Along the way, we generalize Markov numbers to any couple (p, q) ∈ N 2 (not only when they are relatively primes) and conjecture that the unicity is still true as soon as p ≤ q. Finally, we show that the three conjectures are in fact true for this superset.

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“…As we proved Proposition 18 for all couples (d, n) ∈ N 2 , it directly implies the fixed numerator conjecture which was already solved in [11]:…”

Section: Application To the Fixed Denominator And Numerator Conjecturessupporting

confidence: 69%

Section: Introductionmentioning

confidence: 99%

“…The fixed numerator conjecture was proved in [11] using cluster algebra and snake graphs. The proof is quite technical with many interesting tools on discrete paths on the N 2 grid.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Markov numbers: Fixed numerator, denominator, and sum conjectures

Lagisquet

Pelantová

Tavenas

et al. 2021

Advances in Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…In the present paper, all notations and conventions follow those of [3] unless stated explicitly. For a finite word w, we mean by |w| the length of w, i.e., the number of letters appearing in w. For a letter a, let |w| a denote the number of occurrences of a in w. For example, |22211| = 5 and |22211| 2 = 3.…”

Section: Introductionmentioning

confidence: 99%