Conjecture 1.9

Definitions and mathematical statement

A Markov triple is a triple of positive integers satisfying \[ a^2+b^2+c^2=3abc. \] Call it normalized when $1\le a\le b\le c$. The ordering removes permutations of the same coordinates while allowing equal entries. A Markov number is a positive integer occurring as the largest coordinate of at least one normalized Markov triple.

For each positive integer $m$, define \[ \mathcal P(m)=\{(a,b)\in\mathbb Z^2:1\le a\le b\le m, \ a^2+b^2+m^2=3abm\}. \] Is $|\mathcal P(m)|\le1$ for every $m$? Equivalently, whenever $m$ is a Markov number, does its value determine the other two coordinates of the normalized triple uniquely? This is a statement about all positive integer solutions, without a restriction on the size, primality, or factorization of $m$.

The equation is symmetric, so keeping the largest coordinate fixed without an ordering convention would introduce irrelevant exchanges of $a$ and $b$. The proposed uniqueness is exactly uniqueness of the ordered normalized pair. It does not say that a number occurs in only one arbitrary position among all triples.

For a negative answer one must exhibit the same $m$ with two genuinely different pairs in $\mathcal P(m)$. For a positive answer, checking any finite range of $m$ is insufficient: the assertion ranges over the unbounded set of all integers for which such a pair exists.

Short English statement

Does the largest entry of a normalized positive Markov triple uniquely determine the other two entries?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700008 (AMR-086-0008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Waldschmidt - Open Diophantine Problems (2004), Conjecture 1.9. Original source indexed by AMR-086-0008. https://arxiv.org/abs/math/0312440.