Diophantine equations and marked quadratic ideal classes

Markov uniqueness conjecture

Collaboration beta

Can two different positive Markov triples have the same largest number? The packet translates a possible collision into conditions on specifically marked quadratic ideals and into exact integer reconstruction tests. Those reductions do not yet exclude a first proper split for every maximum.

x^{2} + y^{2} + M^{2} = 3 xyM, 1 \leq x \leq y \leq M : M determines (x, y) .

Known results and sources

Illustration: Does the largest entry determine the lower pair of a normalized positive Markov triple?

Research problem

Exact mathematical statement

A normalized Markov triple is a positive integral solution $(x,y,M)$ of $x^2+y^2+M^2=3xyM$ with $x\le y\le M$ . Prove that the largest entry determines the normalized triple:

\begin{gathered} x^2+y^2+M^2=3xyM,\qquad {x'}^2+{y'}^2+M^2=3x'y'M,\\ 1\le x\le y\le M,\quad 1\le x'\le y'\le M \quad\Longrightarrow\quad (x,y)=(x',y'). \end{gathered}

The repeated-coordinate triples $(1,1,1)$ , $(1,1,2)$ , and maximum 5 are elementary cases. the source does not prove the universal implication; a nontrivial collision can be oriented as $1\le a<c<d<b<M$ .

Problem infographic

Problem at a glance

Illustration: Two normalized positive Markov equations share M, above the small triples (1,1,1), (1,1,2), and (1,2,5). — Illustration: Markov uniqueness asks whether two positive normalized solutions with the same largest entry must be the same triple.

Current mathematical picture

Where work on Markov uniqueness conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureReplace marked ideal orders by the presence of an abstract torsion subgroup.

At the actual triple (13,34,1325), the source reports that the marked prime above 5 has narrow and ordinary order 40, whereas a collision through the block 25 would require order 8. Abstract C₈×C₄ is nevertheless present, and only the empty/full marked zero subsets occur. Use exact marked classes from the given triple. The source’s fourth-rank upper bound can prove uniqueness when r₄=1, but larger torsion still requires the marked zero-subset analysis.

Route status · Narrowed route

Main reductionCurrent reduction

Fix an actual normalized triple with maximum M beyond the elementary cases, put η=1 for odd M and η=2 for even M, and write m₀=M/η=∏ℓ_i^{E_i}. In the source’s specified quadratic order, mark 𝔭_i by β_M≡ab⁻¹ mod ℓ_i and let g_i=[𝔭_i^{E_i}]⁺, x_I=∑_{i∈I}g_i. The source gives N(M)=½#{I:4x_I=0}. Hence uniqueness is equivalent to excluding every proper nonempty marked subset with 4x_I=0, retaining the legitimate empty/full boundary.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeExclude every proper nonempty zero subsum of the actual marked sequence 4g_i.Task status · Ready to work on

Work mapped so far

Markov uniqueness conjecture in numbers

1.2kretained lines of mathematical investigation1,181 in the current working snapshot

Argument development: 805 · 68%
Explored or eliminated routes: 59 · 5%
Computational analysis: 51 · 4%
Open obligations: 160 · 14%
Definitions and setup: 106 · 9%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byG. FrobeniusProposed1913Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Exclude every proper nonempty zero subsum of the actual marked sequence 4g_i.

Suggested move: Fix the exact order and marking from an actual triple, partition the complete prime-power blocks, and prove 4∑_{i∈I}g_i≠0 for every proper nonempty I. Preserve the empty/full boundary and address mixed proper subsums, which first become a separate issue with four blocks.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeReplace marked ideal orders by the presence of an abstract torsion subgroup.

Route status · Narrowed route

Narrowed routeSeek a new collision by iterating only the fixed reflection group of an existing collision candidate.

For the source’s fixed collision data, the denominator formula allows integral orbit members only at t=0 with s=−1 or 0. This closes that particular descent route; it does not exclude the initial collision, changes of M, or transformations outside the group. A new transformation must change the marked or parent data, or leave the fixed group with a justified arithmetic lift. Distinct-field square and divisibility conditions remain separate requirements.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Exclude every proper nonempty zero subsum of the actual marked sequence 4g_i.Suggested move: Fix the exact order and marking from an actual triple, partition the complete prime-power blocks, and prove 4∑_{i∈I}g_i≠0 for every proper nonempty I. Preserve the empty/full boundary and address mixed proper subsums, which first become a separate issue with four blocks.

Ready to work on

Exclude all admissible distinct-field two-square points in the collision chamber.Suggested move: Use the exact pair of square conditions with unbounded positive a,n,M and retain 1≤a<c<d<b<M after reconstruction. Common-field height exclusion is already the source’s restricted branch; a new argument must control the remaining biquadratic case rather than repeat a fixed-parameter scan.

Ready to work on

Rule out the positive single-square reconstruction for every proper unitary factor.Suggested move: For each ordered proper factorization in the source’s domain, check whether the specified congruence class has a representative e satisfying 0<e/(p−q)<1. If it does, use that sole representative to form the prescribed Σ and P, and exclude Σ>2P>0 with Ω=(Σ²−4P²)/q² a square. Zero is excluded. Do not add an R<S assumption or resurrect a second divisibility test after the stated reconstruction lemma.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen conjecture

Uniqueness of a positive integral solution of x²+y²+z²=3xyz from its largest entry, up to coordinate permutation, remains open in general. Prime-power cases and alternative formulations are established or reported separately; the 2025 necklace reformulation does not close the full uniqueness conjecture.

[1][2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025-08-22
Peer reviewedFisac expresses the uniqueness question as injectivity of a function on primitive integral necklaces with small variation.[1]
2013
Authoritative summaryAigner's monograph brings together the number-theoretic, geometric, word, and matching perspectives on the Markov theme.[3]
2007-02-01
PreprintZhang's revised preprint gives an elementary argument for uniqueness when a Markoff number is a prime power or twice a prime power.[2]
1913
Authoritative summaryFisac's historical account attributes the uniqueness conjecture to Frobenius and cites his 1913 paper. That original paper was not inspected in this collection.[1]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction1 of 91
lemma3 of 93
equivalence2 of 92
negative result1 of 91
computational claim1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included

retained route statementThe largest entry M determines the normalized Markov triple (x,y,M).
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementMarked subsets count normalized triplesintermediate
retained route statementOne square remains after the canonical residueintermediate
retained route statementPartial powers and whole prime-power blocks differintermediate
retained route statementCommon-field collision bounds are incompatibleintermediate
retained route statementA fixed reflection group yields no new normalized pairintermediate
retained route statementMarked order 40 defeats an abstract-group shortcutintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureReplace marked ideal orders by the presence of an abstract torsion subgroup.reported failure
Useful failureSeek a new collision by iterating only the fixed reflection group of an existing collision candidate.reported failure
Research targetExclude every proper nonempty zero subsum of the actual marked sequence 4g_i.open
Research targetExclude all admissible distinct-field two-square points in the collision chamber.open
Research targetRule out the positive single-square reconstruction for every proper unitary factor.open
ComputationThe source’s order-40 example uses (a,b,M)=(13,34,1325), m₀=25·53, and the prime above 5 marked by β_M≡2 mod 5. The second-audit report separately lists ten exact identities, 384 conic inverses, 205 ideal-product checks, and other bounded controls.The order-40 certificate is a bounded norm-search result for the marked class, not a computation of every class group or a universal uniqueness proof. The earlier 152,209,774-case scan was not rerun in the second audit. · reported unreproduced
Narrowed routeReplace marked ideal orders by the presence of an abstract torsion subgroup.At the actual triple (13,34,1325), the source reports that the marked prime above 5 has narrow and ordinary order 40, whereas a collision through the block 25 would require order 8. Abstract C₈×C₄ is nevertheless present, and only the empty/full marked zero subsets occur. Use exact marked classes from the given triple. The source’s fourth-rank upper bound can prove uniqueness when r₄=1, but larger torsion still requires the marked zero-subset analysis.
Narrowed routeSeek a new collision by iterating only the fixed reflection group of an existing collision candidate.For the source’s fixed collision data, the denominator formula allows integral orbit members only at t=0 with s=−1 or 0. This closes that particular descent route; it does not exclude the initial collision, changes of M, or transformations outside the group. A new transformation must change the marked or parent data, or leave the fixed group with a justified arithmetic lift. Distinct-field square and divisibility conditions remain separate requirements.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeExclude every proper nonempty zero subsum of the actual marked sequence 4g_i.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointExclude every proper nonempty zero subsum of the actual marked sequence 4g_i.

Markov uniqueness conjecture · ready to start

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Research contextPrepared context for any AI agent

Can two different positive Markov triples have the same largest number? The current work translates a possible collision into conditions on specifically marked quadratic ideals and into exact integer reconstruction tests. Those reductions do not yet exclude a first proper split for every maximum.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Markov’s conjecture on integral necklacespeer reviewed result · David Fisac · Bulletin of the London Mathematical Society 57, 4122–4131 · 2025-08-22 · DOI 10.1112/blms.70178 · accessed Sep 7, 2026
2
An elementary proof of uniqueness of Markoff numbers which are prime powerspreprint · Ying Zhang · arXiv · 2007-02-01 · ARXIV math/0606283 · accessed Sep 7, 2026
3
Markov's Theorem and 100 Years of the Uniqueness Conjecturesurvey or monograph · Martin Aigner · Springer · 2013 · DOI 10.1007/978-3-319-00888-2 · accessed Sep 7, 2026

Important qualifications

This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
The 1913 attribution is supported by Fisac's historical account; it does not identify a packet author, priority claim, or a directly inspected original source.
MSC 11D45 appears on the cited Zhang preprint. No exact checked formalization or computational certificate was confirmed.

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Markov uniqueness conjecture

Exact mathematical statement

Problem at a glance

Where work on Markov uniqueness conjecture stands

Markov uniqueness conjecture in numbers

Exclude every proper nonempty zero subsum of the actual marked sequence 4g_i.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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