Arithmetic dynamics of divisor sums

Catalan–Dickson aliquot conjecture

Collaboration beta

Repeatedly replace a positive integer by the sum of its proper divisors. Must every sequence eventually repeat or reach zero? The source rules out several exact descriptions of escape and strengthens a specific counterexample route, while arbitrary orbits remain uncontrolled.

\forall n_{0} \geq 1 \exists t_{0} \geq 0, ℓ \geq 1 \forall t \geq t_{0} : n_{t + ℓ} = n_{t}, n_{t + 1} = s (n_{t})

Known results and sources

Illustration: Repeated proper-divisor sums can reach zero or repeat. Must every positive-integer start eventually repeat?

Research problem

Exact mathematical statement

For each integer $n_0\ge1$ , define the aliquot sequence by

n_{t+1}=s(n_t),\qquad s(n)=\sum_{\substack{d\mid n\\1\le d<n}}d=\sigma(n)-n\ (n\ge1),\qquad s(0)=0.

Prove

\forall n_0\in\mathbb Z_{\ge1}\ \exists t_0\in\mathbb Z_{\ge0}\ \exists\ell\in\mathbb Z_{\ge1}\ \forall t\ge t_0: \quad n_{t+\ell}=n_t.

Absorbing zero, perfect-number fixed points, amicable pairs, and cycles of any length are included. The conjecture is equivalent to boundedness of every orbit, not to termination of every orbit or a list of known cycles. A counterexample requires one ordinary positive integer with a nonrepeating trajectory that never reaches zero. Such an orbit tends to infinity but need not increase at every step.

Problem infographic

Problem at a glance

Illustration: An exact finite aliquot chain and the fixed point 6 accompany the divisor-sum definition and the question of eventual periodicity for every positive start. — Illustration: The conjecture includes absorbing zero and positive cycles. A finite example does not settle all positive-integer starts.

Current mathematical picture

Where work on Catalan–Dickson aliquot conjecture stands

Open conjecture

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

Useful failurePromoting a finite-filter symbolic path to an ordinary prime orbit

The UV U²V U³V… construction is compatible with the finite filters and unique-marker restriction; its corrected long-gap argument uses k≥2. An input-dependent arithmetic implication controlling integrality and prime values for one ordinary start could still settle R-AB.

Route status · Narrowed route

Main reductionOne ordinary legal return itinerary would refute the target

For an ordinary prime p>11, A(p)=(37p+72)/35 is legal only when A(p)>11 is prime; B(p)=(1961p+9756)/1925 is legal only when both Q(p)=(37p+72)/55 and B(p) exceed 11 and are prime. An infinite legal itinerary would give an escaping actual orbit with squarefree Ω=4 terms avoiding 3. Such a start is not proved to exist, and leaving the subsystem is not termination.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.Task status · Ready to work on

Work mapped so far

Catalan–Dickson aliquot conjecture in numbers

1.2kretained lines of mathematical investigation1,176 in the current working snapshot

Argument development: 823 · 70%
Explored or eliminated routes: 48 · 4%
Computational analysis: 33 · 3%
Open obligations: 144 · 12%
Definitions and setup: 128 · 11%

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 byE. Catalan, L. E. DicksonStatus checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.

Suggested move: Suggested next step: Formulate an invariant on one actual orbit that bounds every prime factor of both members of selected adjacent pairs, and prove it without assuming bounded Ω, abundancy, or a fixed driver.

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 routePromoting a finite-filter symbolic path to an ordinary prime orbit

Route status · Narrowed route

Narrowed routeSelecting an ordinary integer by profinite unit tests

The initial coordinate equals −271 in Z_5 and −270 in Z_181, so no ordinary integer has both exact values. The construction asserts polynomial identities rather than a globally defined division map. Arguments choosing a modulus from a particular ordinary input remain possible; neither CRT approximation nor profinite compactness alone supplies simultaneous primality.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.Suggested move: Suggested next step: Formulate an invariant on one actual orbit that bounds every prime factor of both members of selected adjacent pairs, and prove it without assuming bounded Ω, abundancy, or a fixed driver.

Ready to work on

Resolve ordinary-input realizability in the infinite legal 70/110 subsystem while distinguishing that route from the full target.Suggested move: Suggested next step: Develop a switch-sensitive obstruction or exhibit one fixed ordinary start, checking every A-output and both Q and B prime outputs and testing the symbolic and profinite negative controls.

Ready to work on

Extend finite prime-exponent preservation to a changing sequence that actually belongs to the same intended aliquot orbit.Suggested move: Suggested next step: Specify a nonperiodic class beyond CD-A09, prove orbit membership, and show that finitely many forced prime exponents suffice while every refinement preserves earlier conditions.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen conjecture

The selected sources leave open whether every positive-integer aliquot sequence reaches 1 or becomes periodic. Finite iteration and factorization records do not rule out an unbounded sequence.

[1][2]

External progress

What the literature has established

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

2018
Peer reviewedBosma investigated starts below one million, iterating until termination, a cycle, or a 100-digit term. Reaching that cutoff does not settle the sequence's eventual behavior.[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
reduction2 of 92
lemma4 of 94
equivalence1 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.24 displayed rows · 2 routes included

retained route statementMust every repeated proper-divisor sum eventually repeat?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementAdjacent prime-support completion criterionintermediate
retained route statementExact periodic rational schedules cannot escapeintermediate
retained route statementExact quasipolynomial trajectories cannot escapeintermediate
retained route statementOne ordinary legal return itinerary would refute the targetintermediate
retained route statementEvery legal infinite itinerary needs unique markersintermediate
retained route statementFour-prime certificate imposes high word complexityintermediate
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 failurePromoting a finite-filter symbolic path to an ordinary prime orbitreported failure
Useful failureSelecting an ordinary integer by profinite unit testsreported failure
Research targetProve actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.open
Research targetResolve ordinary-input realizability in the infinite legal 70/110 subsystem while distinguishing that route from the full target.open
Research targetExtend finite prime-exponent preservation to a changing sequence that actually belongs to the same intended aliquot orbit.open
Narrowed routePromoting a finite-filter symbolic path to an ordinary prime orbitThe UV U²V U³V… construction is compatible with the finite filters and unique-marker restriction; its corrected long-gap argument uses k≥2. An input-dependent arithmetic implication controlling integrality and prime values for one ordinary start could still settle R-AB.
Narrowed routeSelecting an ordinary integer by profinite unit testsThe initial coordinate equals −271 in Z_5 and −270 in Z_181, so no ordinary integer has both exact values. The construction asserts polynomial identities rather than a globally defined division map. Arguments choosing a modulus from a particular ordinary input remain possible; neither CRT approximation nor profinite compactness alone supplies simultaneous primality.

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 bridgeProve actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.

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.

Continue the mathematics

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Prepared starting pointProve actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.

Catalan–Dickson aliquot conjecture · ready to start

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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
Aliquot sequences with small starting valuespeer reviewed result · Wieb Bosma · INTEGERS 18, A75 · 2018-08-31 · ARXIV 1604.03004 · accessed Sep 7, 2026
2
Aliquot sequencesauthoritative webpage · Wieb Bosma · accessed Sep 7, 2026
3
Catalan–Dickson Lean conjecture statement, pull request #2403formalization · google-deepmind/formal-conjectures · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.

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Catalan–Dickson aliquot conjecture

Exact mathematical statement

Problem at a glance

Where work on Catalan–Dickson aliquot conjecture stands

Catalan–Dickson aliquot conjecture in numbers

Prove actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.

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

Formal and computational footholds

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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