ProofAtlas

Multiplicative number theory

Odd Perfect Number Conjecture

Collaboration beta

A perfect number equals the sum of its proper divisors; the conjecture says that no odd integer can have this property.

Nodd⇒σ(N)≠2N
Known results and sources
A patterned odd-number factor constellation faces a balance between one number and its divisor sum, separated by an open question.
The card visualizes the definition of perfection and leaves the possible odd case unresolved, without showing the packet's factor-graph route.

Research problem

Exact mathematical statement

Prove that no odd perfect number exists. Equivalently, if σ(N)\sigma(N) denotes the sum of the positive divisors of NN, prove

Nodd⇒σ(N)≠2N.N\text{ odd}\quad\Longrightarrow\quad\sigma(N)\ne 2N.

The retained source explicitly says that the active frontier remains open and does not claim a proof.

Problem infographic

Problem at a glance

A three-panel explainer defines the divisor-sum condition for perfect numbers and asks whether one can prove that no odd perfect number exists.
The plate states the odd perfect number conjecture as an open nonexistence claim and avoids presenting source-reported bounds as a proof.

Current mathematical picture

Where work on Odd Perfect Number Conjecture stands

Open conjecture

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

Useful failureRepeating retired finite branch censuses

The source's recorded limitation records those cases as exhausted while leaving the arbitrary-support and uniform wide-gap problems open. Seek a genuinely uniform valuation, carry, or multiplicative-potential obstruction and audit the symbolic bridges independently.

Route status · Narrowed route
Main reductionEuler-form setup

The source assumes an odd perfect number in Euler form and introduces S and h to expose divisor-sum structure.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeDerive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters.Task status · Ready to work on
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

Odd Perfect Number Conjecture in numbers

3kretained lines of mathematical investigation2,987 in the current working snapshot
Argument development
2,595 · 87%
Explored or eliminated routes
34 · 1%
Computational analysis
153 · 5%
Open obligations
106 · 4%
Definitions and setup
99 · 3%
6selected mapped statements1routes 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.

Status checkedAug 15, 2026
Source recordedAug 15, 2026

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

10 selected steps

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

10 selected steps

Scroll horizontally to explore the route

Working route overview for Odd Perfect Number ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can an odd integer equal the sum of its proper divisors? — Depends on missing premiseCan an odd integer equal thesum of its proper divisors?Current reduction — Depends on missing premiseCurrent reductionEuler-form setup — Depends on missing premiseEuler-form setupClosing target — Depends on missing premiseClosing targetConditional symbolic reductions — Depends on missing premiseConditional symbolicreductionsProject-audited h bound — Depends on missing premiseProject-audited h boundRepeating retired finite branch censuses — stoppedRepeating retired finitebranch censusesDerive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters. — OpenDerive a uniform obstructionfor the Type IIlinear-return…Prove a uniform wide-gap carry theorem whose lift-depth bound grows with the normalized gap. — OpenProve a uniform wide-gapcarry theorem whoselift-depth…Control arbitrary factor support with a global graph or multiplicative-potential argument. — OpenControl arbitrary factorsupport with a global graphor…
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

1 recorded
Narrowed routeRepeating retired finite branch censuses

The source's recorded limitation records those cases as exhausted while leaving the arbitrary-support and uniform wide-gap problems open. Seek a genuinely uniform valuation, carry, or multiplicative-potential obstruction and audit the symbolic bridges independently.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Derive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters.Suggested move: Use congruences, Pell or Thue reductions for fixed small sources, and the exact auxiliary identity to isolate a parameter-independent contradiction.
Ready to work on
02
Prove a uniform wide-gap carry theorem whose lift-depth bound grows with the normalized gap.Suggested move: Extract a monotone lower bound from the exact carry recursion and its tail, covering every gap beyond the retained finite certificate.
Ready to work on
03
Control arbitrary factor support with a global graph or multiplicative-potential argument.Suggested move: Apply the generalized cut inequality to strongly connected support components and seek a strictly increasing but globally bounded potential.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

No odd perfect number is known, and the existence question remains open. Current literature continues to derive necessary factor and exponent conditions for a hypothetical example rather than a complete existence or nonexistence theorem.

[1][3]
External progress

What the literature has established

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

  1. PreprintOchem and Zelinsky derived a strong divisibility condition under the special hypothesis that exactly one even exponent exceeds two.[3]
  2. Peer reviewedNielsen proved that a hypothetical odd perfect number has at least ten distinct prime divisors, using Diophantine bounds and computation.[2]
  3. Peer reviewedHare proved that the total number of prime factors of a hypothetical odd perfect number, counted with multiplicity, is at least 47.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusOdd Perfect Number Conjecture
Dependency or reductionprime-factor lower bounds for odd perfect numbers

Lower bounds on the number of distinct prime divisors restrict any hypothetical example but do not decide existence.

[2]
Weaker or relaxed formone even exponent greater than two

The one-exception exponent pattern is a restricted hypothetical family within the full odd perfect number problem.

[3]

Formal and computational footholds

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

  • computation · not independently reproducedNielsen distinct-prime-factor computation

    The peer-reviewed paper uses extensive computations to derive at least ten distinct prime factors; those computations were not rerun here.

    [2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetFormal divisor-sum arithmetic, perfect-number definitions, and Euler's structural form for odd perfect numbers.
  • Formalization targetChecked cyclotomic, primitive-divisor, valuation, and Diophantine ingredients used by modern necessary-condition results.
  • Formalization targetReproducible certificate formats for the computation-assisted factor-count bounds.

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
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.

Detailed research inventory

Claims, milestones, and routes in the current map

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

4 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction2 of 62
  • lemma3 of 63
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementCan an odd integer equal the sum of its proper divisors?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementEuler-form setupintermediate
  • retained route statementProject-audited h boundintermediate
  • retained route statementConditional symbolic reductionsintermediate
  • 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
  • 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 failureRepeating retired finite branch censusesreported failure
  • Research targetDerive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters.open
  • Research targetProve a uniform wide-gap carry theorem whose lift-depth bound grows with the normalized gap.open
  • Research targetControl arbitrary factor support with a global graph or multiplicative-potential argument.open
  • Research targetNonexistence of odd perfect numberssuperseded
  • Research targetUniform wide-gap carrysuperseded
  • Research targetArbitrary supportsuperseded
  • Narrowed routeRepeating retired finite branch censusesThe source's recorded limitation records those cases as exhausted while leaving the arbitrary-support and uniform wide-gap problems open. Seek a genuinely uniform valuation, carry, or multiplicative-potential obstruction and audit the symbolic bridges independently.
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 bridgeDerive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters.

1 approach has 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.

Continue the mathematics

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Prepared starting pointDerive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters.

Odd Perfect Number Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

A perfect number equals the sum of its proper divisors; the conjecture says that no odd integer can have this property.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
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Sources and references3 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    More on the total number of prime factors of an odd perfect numberpeer reviewed result · Kevin G. Hare · Mathematics of Computation · 2005 · DOI 10.1090/S0025-5718-04-01683-7 · accessed Aug 15, 2026
  2. 2
    Odd perfect numbers, Diophantine equations, and upper boundspeer reviewed result · Pace P. Nielsen · Mathematics of Computation · 2015 · DOI 10.1090/S0025-5718-2015-02941-X · accessed Aug 15, 2026
  3. 3
    On odd perfect numbers with exactly one even exponent greater than 2preprint · Pascal Ochem, Joshua Zelinsky · arXiv · 2026-07-22 · ARXIV 2607.19746 · accessed Aug 15, 2026

Important qualifications

  • No existence or nonexistence claim is inferred from necessary conditions on a hypothetical odd perfect number.
  • The 2026 preprint is a conditional exponent-pattern result and is not presented as a resolution.
  • The AMS papers support established lower bounds and the no-known-example status; literature coverage is not exhaustive.
  • No packet URL or attachment was fetched, executed, rendered, or treated as external evidence.

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