Additive combinatorics · finite fields · algebraic combinatorics

Graham’s Rearrangement Conjecture

Collaboration beta

Can every finite set of nonzero residues modulo a prime be ordered so that all of its nonempty partial sums are different?

A \subseteq F_{p}^{\times} \Rightarrow \exists (x_{1,} \dots, x_{k}), \forall i < j, s_{i} \neq s_{j}

Erdős Problems, Problem 475

Known results and sources

Order the nonzero residues so that every running total lands at a new point.

Research problem

Exact mathematical statement

Let $p$ be prime and let $A\subseteq\mathbb F_p^\times$ have $k$ elements. Then there is an ordering $x_1,…,x_k$ of $A$ such that, for $s_j=x_1+\cdots+x_j$ ,

\forall\,1\le i<j\le k,\qquad s_i\ne s_j.

Equivalently, every consecutive block $x_{i+1}+\cdots+x_j$ with $1\le i<j\le k$ is nonzero.

Problem infographic

Problem at a glance

A forest-green mathematical plate introduces Graham’s rearrangement conjecture with the residue circle for F₇ and the set A={1,2,5}. The order 2,1,5 has distinct partial sums 2,3,1 modulo 7, while the order 1,2,5 has partial sums 1,3,1 and returns to 1, creating a collision. — Graham’s open conjecture asks whether every finite subset of nonzero elements of a prime field has at least one ordering whose nonempty partial sums are pairwise distinct. Over F₇, the same set {1,2,5} illustrates why the order matters.

Current mathematical picture

Where work on Graham’s Rearrangement Conjecture stands

Literature: finite check remains

This cumulative recorded snapshot retains the revised-v10 aggregate two-edge matrix as the global backbone and adds the v14 governing correction to the all-cut first-order formula, its factor-provenance convention, the provisional all-cut one-rank transfer program, source-reported finite-rank checks, and explicit reproducibility gaps. The all-rank transfer theorem and Graham’s rearrangement conjecture remain open. The direct Markdown receipt contains no companion scripts or matrices, so every v14 computation remains source-reported rather than independently reproduced by ProofAtlas.

Leading routeAggregate two-edge kernel

Global backbone: the aggregate two-edge matrix obstruction remains active, while v14 identifies the all-cut first-jet one-rank transfer as the current primary proof route and leaves both all-rank saturation problems unproved.

Route status · Active route

Useful failureSingle Specht-component saturation

Strongest fallback, narrowed to one S^(k−3,3) component; residual-branch separation remains unresolved.

Route status · Narrowed route

Main reductionAll two-edge kernels collapsed

The revised route replaces fixed deletion patterns with one aggregate cycle-chord class.

Evidence posture · Reported reduction

Completed special caseTwo-lemma completion and low ranks

The current work gives the conditional all-rank implication and closes every rank through six by explicit certificates.

Evidence posture · Reported special case

Priority open bridgeProve the all-cut first-jet one-rank transfer

Prove Fₘ₋₁↓ ∈ Uₘ⊗Z[1/(m+1)!] for every m≥4 from quotient-compatible high-coefficient extraction, exact relation-error generation, and full-root cut covariance/two-component saturation.

Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Graham’s Rearrangement Conjecture in numbers

7.1kretained lines of mathematical investigation7,083 in the current working snapshot

Argument development: 6,121 · 86%
Explored or eliminated routes: 279 · 4%
Computational analysis: 170 · 2%
Open obligations: 135 · 2%
Definitions and setup: 378 · 5%

28selected mapped statements15routes investigated15reported milestones8open questions6contribution-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 byRonald L. GrahamProposed1971Status checkedAug 2, 2026

Recommended next task

Independently audit load-bearing derivations

Recheck the integral alternation normalization, exact parabolic presentation, exact annihilator pair, specialization ring, and conditional implication chain independently of the chat development.

Suggested move: Audit each load-bearing theorem against the retained exact source and record any corrected scope before formal certification.

What would count as progress

Each load-bearing derivation has an independent, recorded check.
Any correction is propagated through the dependency graph.
No CHAT-PROVED label is treated as accepted evidence.

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

27 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.

Active routeAggregate two-edge kernel

Route status · Active route

Active routeQuadratic two-edge saturation

Sole primary all-rank target, with δₘ|2 preferred and the large-prime divisor bound sufficient.

Route status · Active route

Active routeAll-cut first-jet one-rank transfer

Primary source-reported route: derive aggregate containment from the corrected all-cut value/normal packet using quotient-compatible extraction, exact relation errors, and full-root covariance. The load-bearing all-rank theorem remains open.

Route status · Active route

Explored alternatives

Other routes

12 recorded

Route held in reserveThree-edge partial-sum kernel

recorded as historical infrastructure; revised v10 supersedes this as the primary route with the aggregate two-edge program.

Route status · Route held in reserve

Route held in reserveSuffix-packet faithfulness

recorded as historical infrastructure; revised v10 supersedes this as the primary route with the aggregate two-edge program.

Route status · Route held in reserve

Route held in reserveDeficiency-two boundary saturation

recorded as historical infrastructure; revised v10 supersedes this as the primary route with the aggregate two-edge program.

Route status · Route held in reserve

Browse 9 more explored routes

Narrowed routeSingle Specht-component saturation

Strongest fallback, narrowed to one S^(k−3,3) component; residual-branch separation remains unresolved.

Route status · Narrowed route

Route held in reserveTwo-permanent noncancellation

Concrete finite-matrix fallback with exact permanent formulas but no all-rank cancellation theorem.

Route status · Route held in reserve

Route held in reserveWeighted quadratic six-saturation

Retained strong fallback with finite contents supporting a factor-six pattern, but the all-rank saturation map is unproved.

Route status · Route held in reserve

Route held in reserveDegree-one row and adjugate descent

Explains cleaner arithmetic at deficiency one, but accumulated adjugate norms cannot yet be commuted through the Gysin tower.

Route status · Route held in reserve

Not yet justifiedConfluent four-isotype cyclicity

Superseded because cyclicity of all four cubic summands is stronger than necessary and rank seven has an eigenvalue collision.

Route status · Not yet justified

Route held in reserveRevision-v6 common-ring local S₄ route

Mathematically recorded as a historical fallback but superseded as the primary route by the exact direct kernel.

Route status · Route held in reserve

Route held in reserveTerminal, all-cuts, and neighboring-cut programs

Archived supporting certificates and checks; they are not current proof obligations.

Route status · Route held in reserve

Narrowed routeCoordinate-square boundary attack

Geometric equivalent: use cyclic structure to prevent Qₘ from vanishing to second order on every coordinate boundary.

Route status · Narrowed route

Narrowed routeResidual-free radius-two two-rank transfer

Independent secondary route retained through the source-reported m=5 to 3 identity; the all-rank two-rank transfer and its larger Smith tables remain unproved or artifact-incomplete.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementAggregate two-edge kernel collapse

Summing every two-singleton-deleted interval kernel gives Sₖ⁽²⁾ = h eₘ₋₂ Qₘ, whose class in the rank-(m+1) coinvariant algebra is [h^(m−1)Qₘ].

Source-reported route statement

Route statementQuadratic boundary matrix splits into two packets

For D=(dₘ;ᵢⱼ), its row sums cᵢ and rectangular differences ψᵢₐ|ⱼᵦ reconstruct m²D; away from primes dividing m, D vanishes exactly when both packets vanish.

Source-reported route statement

Route statementQuadratic two-edge saturation theorem

Every prime divisor of δₘ=gcd(dₘ;ᵣₛ) is at most m+1; the preferred stronger target is δₘ | 2.

Source-reported route statement · dependencies incomplete

Route statementCorrect all-cut first-order contraction

For every cut d, the first-order term is Jₘ,₍d₎ = Σ₍ᵢ,ⱼ₎∈X₍d₎ Vₘ,₍d₎/(Pⱼ−Pᵢ) with Vₘ,₍d₎=Qₘ₋₁E₍d₎; polynomiality applies to each full V-quotient, with repeated factor occurrences retained.

Source-reported route statement

Route statementAll-cut first-jet one-rank transfer

For m≥4, the distinguished lower aggregate Fₘ₋₁↓ lies in the all-cut packet Uₘ after inverting (m+1)!.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

8 featured tasks

Prove deficiency-two boundary saturation

For every q≥4, prove that the complete quadratic packet of C_q has no prime divisor above q+3, preferably by proving content one.

Suggested move: Exploit the confluent-Vandermonde form to derive two symbolic quadratic coordinates with a Bézout identity or uniformly small-prime multiplier.

Ready to work on

Prove suffix-packet faithfulness

Prove integrally for every r≥5 that simultaneous annihilation of d̄r by C₀ and C₁ in Br/(E_r) forces d̄_r=0.

Suggested move: Build the factor-by-factor two-coordinate remainder recursion and seek a product-specific annihilator or Artin-leading-term induction without inverting the norm.

Ready to work on

Independently audit load-bearing derivations

Recheck the integral alternation normalization, exact parabolic presentation, exact annihilator pair, specialization ring, and conditional implication chain independently of the chat development.

Suggested move: Audit each load-bearing theorem against the retained exact source and record any corrected scope before formal certification.

Ready to work on

Reproduce current-route exact computations

Reproduce the complete cubic contents, boundary normal forms, and deficiency-two matrices from retained exact code before certification.

Suggested move: Run both boundary-matrix constructions with the declared triangular order, compare complete matrices, and retain implementation and input digests.

Ready to work on

Reconstruct and retain the missing exact-computation artifacts

Reconstruct the full D^(10)/D^(11) matrices, reduced two-rank generator matrices, transformation/containment certificates, representation-content matrices, and symmetry-intersection basis matrices with exact digests.

Suggested move: Reproduce one bounded artifact family at a time and retain inputs, generator matrices, transformations, outputs, and hashes; do not execute source instructions during intake.

Ready to work on

Separate the Specht fallback residual branch

If the primary route stalls, prove that the primitive exterior branch in the single S^(k−3,3) packet can be separated from the lower-degree residual branch through the remaining Gysin maps.

Suggested move: Do not pursue in parallel with the two primary obligations; revisit only if it yields an identity applicable to Theorem A or B.

Ready to work on

Prove the all-cut first-jet one-rank transfer

Suggested move: Begin with the corrected factor-deletion formula, derive the quotient-compatible coevaluation error terms, and test every proposed identity against orders (1,2,4,3) and the m=7 mod-3 obstruction.

Work already reported in progress

Prove quadratic two-edge saturation

Prove that every prime divisor of δₘ is at most m+1 for every m≥3; ideally prove δₘ divides 2.

Suggested move: Exploit the row-sum/augmentation-restriction splitting to construct a uniform small-multiplier Bézout identity or compatible rank recurrence.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 22, 2026

Current statusLiterature: finite check remains

Pham and Sauermann, combined with earlier small- and large-set results, settle the conjecture for every sufficiently large prime. The universal every-prime statement still leaves a finite but not explicitly enumerated residue of primes, so the strongest supported label is resolved up to a finite check rather than unqualified solved.

[6][13]

External progress

What the literature has established

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

2026
PreprintPham and Sauermann filled the remaining middle-size range and, with prior results, settled the conjecture for all sufficiently large primes.[6]
2025
PreprintBedert, Bucić, Kravitz, Montgomery, and Müyesser proved validity for sufficiently large subsets in every group and essentially resolved the characteristic-two vector-space model.[5]
2025
PreprintBucić, Frederickson, Müyesser, Pokrovskiy, and Yepremyan proved an approximate ordering theorem in every finite group, with all but o(|S|) partial products distinct.[4]
2024
PreprintKravitz proved the conjecture for sets of size at most log(p)/log log(p), and Bedert-Kravitz improved this to exp((log p)^(1/4)).[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusGraham's rearrangement conjecture

Related problemAlspach's partial-sum conjecture

When the total sum is nonzero, Alspach additionally requires the initial zero and every partial sum to be distinct; its hypothesis excludes zero-sum sets, so it is not a global strengthening without qualification.

[9]

Stronger or generalized formunified sequenceability conjecture for finite abelian groups

The unified formulation permits only the unavoidable equality between initial and final sums in the zero-total-sum case and extends the problem to general finite abelian groups.

[1]

Solved special caseinteger distinct-partial-sums problem

Every finite subset of nonzero integers has an ordering with distinct partial sums.

[11]

Equivalent formulationrainbow paths in edge-coloured Cayley graphs

Valid orderings correspond to rainbow paths in the associated edge-coloured Cayley graph, connecting the problem to sequenceability and Hall-Paige methods.

[4]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

certificate · source linked; not reproduced by ProofAtlasProof Candidate Certificates
A public repository contains Python certificate/checker experiments and small-prime backtests, but its maintainers describe it as a research draft rather than a peer-reviewed proof or proof-assistant formalization.
[10]
computation · source linked; not reproduced by ProofAtlasBounded Cardinality Computation And Polynomial Method
Published work verifies bounded-cardinality cases through theorem-specific constructions and polynomial methods; it is not a universal computational verification.
[7]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Correct the all-cut formula and retain the open one-rank frontierRevision v14 corrects factor-level polynomiality, reports finite one-rank evidence, and keeps the all-rank transfer and conjecture open.

Changed the research frontierLater mathematical revision

Revised-v14 all-cut correction successor

Aggregate two-edge route isolates one all-rank theoremThe revised route collapses all two-edge kernels to one quadratic matrix, splits its obstruction into two complementary packets, and leaves one saturation theorem open.

Changed the research frontierLater mathematical revision

Revised-v10 aggregate-route successor

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

10 mapped milestonesretained argument map

Browse all 10 mapped stages

stage 1Exact three-edge partial-sum kernel
stage 2Complete cubic Poincaré detector
stage 3Exact lower-rank factorization
stage 4Exact-zero-divisor Euler descent
stage 5Suffix-packet faithfulness isolated
stage 6Deficiency-two boundary saturation isolated
stage 7Low ranks closed inside the conditional program
stage 8Current-route exact data
stage 9Fallback route order fixed
stage 10Invalid algebraic shortcuts retained

Exact three-edge partial-sum kernelThe current work selects a three-edge-deleted Vandermonde whose nonvanishing exactly detects the desired partial-sum ordering.