Wikipedia number-theory item 109: The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exact…

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A normalized Markov triple is $(a,b,c)\in\mathbb N^3$ with $a\le b\le c$ and $a^2+b^2+c^2=3abc$. A Markov number is any component of a positive integer solution of this equation; the mutation $(a,b,c)\mapsto(a,b,3ab-c)$ and permutations show that every Markov number occurs as a largest component. Normalization removes permutations, not triples with equal entries.

Statement recorded in the supplied source.

For every Markov number $c$, is there exactly one pair $(a,b)$ with $1\le a\le b\le c$ and $a^2+b^2+c^2=3abc$? [S2]

Short English statement

Does the largest entry uniquely determine a positive Markov triple once the entries are sorted?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400109 (AMR-093-0109). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Yasuaki Gyoda, An Introduction to the Lagrange and Markov Spectra through the Lens of Generalized Markov Numbers, arXiv:2609.28097v1 (2026) Chapter 1, Markov equation and Conjecture 1.1.20. https://arxiv.org/pdf/2609.28097.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Wikipedia number-theory item 109: The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exact…. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.