ProofAtlas

Decimal digit dynamics and smooth numbers

Multiplicative persistence conjecture

Collaboration beta

Multiply a number’s decimal digits and repeat until one digit remains. The conjecture says this always takes at most eleven steps. The packet classifies seventeen particular inverse problems and proves a finite bound for predecessors of each fixed positive target, while the depth of all possible trajectories remains open.

P(n)=min{r≥0:Dr(n)<10},P(n)≤11.
Known results and sources
Illustration: The exact digit-product chain 39→27→14→4 has three arrows, alongside the open bound P(n)≤11.
Illustration: The chain 39→27→14→4 takes three steps; the conjecture asks whether eleven always suffice.

Research problem

Exact mathematical statement

Let D(n)D(n) be the product of the decimal digits of a nonnegative integer nn, with D(0)=0D(0)=0. Define its multiplicative persistence by

P(n)=min{r≥0:Dr(n)<10}.P(n)=\min\{r\ge0:D^{\,r}(n)<10\}.

One-digit integers have persistence zero. The decimal multiplicative persistence conjecture is

∀n∈ℤ≥0,P(n)≤11.\forall n\in\mathbb Z_{\ge0},\qquad P(n)\le11.

The source supplies an eleven-step example but leaves the universal upper bound open. Strict descent for multidigit integers proves termination, not a uniform bound on the number of steps.

Problem infographic

Problem at a glance

Illustration: A three-step decimal chain sits above the definitions of digit product and persistence, with one-digit and zero-digit conventions.
Illustration: Individual digit-product chains terminate at one digit; the universal eleven-step bound remains open.

Current mathematical picture

Where work on Multiplicative persistence conjecture stands

Open conjecture

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

Useful failureForce every smooth zero-terminal trajectory of persistence at least five through 20.

The source gives 279988627968→2633637888→6967296→244944→4608→0. The starting value is smooth and has persistence five, so the example defeats the threshold-five statement. It does not prove the proposed threshold-six statement. A threshold-six forcing theorem remains an open candidate. A direct uniform bound above all of Z avoids needing that strengthening.

Route status · Narrowed route
Main reductionCurrent reduction

Let S={2^a3^b5^c7^d:a,b,c,d≥0}, and define I(t)={x∈S:x≥10,D(x)=t}. The universal P(n)≤11 bound is equivalent to P(s)≤10 for every s∈S. Using the imported odd-terminal theorem, the remaining endpoints are I^10(Z)=∅ for Z={z∈S:z≥10,D(z)=0}, and I^11({2,4,6,8})=∅. Z is infinite; positive fixed-target height bounds do not make it a finite root set.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve I^10(Z)=∅ for the entire infinite positive zero boundary Z.Task status · Ready to work on

Work mapped so far

Multiplicative persistence conjecture in numbers

1kretained lines of mathematical investigation1,039 in the current working snapshot
Argument development
621 · 60%
Explored or eliminated routes
53 · 5%
Computational analysis
168 · 16%
Open obligations
154 · 15%
Definitions and setup
43 · 4%
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 byN. J. A. Sloane (bounded-persistence conjecture)Status checkedSep 7, 2026
Source recordedSep 7, 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Multiplicative persistence conjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.P(n)≤11 for every n≥0, where P counts digit-product steps until one digit remains. — Depends on missing premiseP(n)≤11 for every n≥0, whereP counts digit-product stepsuntil…A positive first product reduces the target to smooth numbers — Depends on missing premiseA positive first productreduces the target to smoothnumbersCurrent reduction — Depends on missing premiseCurrent reductionThe zero and nonzero-even branches have different indices — Depends on missing premiseThe zero and nonzero-evenbranches have differentindicesAn explicit trajectory attains eleven steps — Depends on missing premiseAn explicit trajectoryattains eleven stepsClosing target — Depends on missing premiseClosing targetEvery fixed positive inverse problem is finite — Depends on missing premiseEvery fixed positive inverseproblem is finiteSeventeen fixed targets have reported complete predecessor sets — Depends on missing premiseSeventeen fixed targets havereported completepredecessor…The inverse tree above 20 has a defined next frontier — Depends on missing premiseThe inverse tree above 20has a defined next frontierForce every smooth zero-terminal trajectory of persistence at least five through 20. — stoppedForce every smoothzero-terminal trajectory ofpersistence…Infer actual smooth trajectories from a fixed-modulus relaxation that inspects only bounded leading and trailing digits. — stoppedInfer actual smoothtrajectories from afixed-modulus…Prove I^10(Z)=∅ for the entire infinite positive zero boundary Z. — OpenProve I^10(Z)=∅ for theentire infinite positivezero…Prove I^11({2,4,6,8})=∅ with every inverse layer fully accounted for. — OpenProve I^11({2,4,6,8})=∅ withevery inverse layer fullyaccounted…Close the conditional route through 20 beyond its first three known inverse layers. — OpenClose the conditional routethrough 20 beyond its firstthree…
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 routeForce every smooth zero-terminal trajectory of persistence at least five through 20.

The source gives 279988627968→2633637888→6967296→244944→4608→0. The starting value is smooth and has persistence five, so the example defeats the threshold-five statement. It does not prove the proposed threshold-six statement. A threshold-six forcing theorem remains an open candidate. A direct uniform bound above all of Z avoids needing that strengthening.

Route status · Narrowed route
Narrowed routeInfer actual smooth trajectories from a fixed-modulus relaxation that inspects only bounded leading and trailing digits.

The current work constructs safe inspected ends for arbitrarily large powers of two and relaxed edges witnessed by other decimal numbers with matching ends, length, and residue. Those witnesses need not be the smooth state itself, so D(s_i)=s_{i+1} can fail throughout an arbitrarily long relaxed path. Adaptive moduli, target-dependent bounds, and whole-word arguments remain available. The limited obstruction does not rule out every finite-state or modular proof.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove I^10(Z)=∅ for the entire infinite positive zero boundary Z.Suggested move: Define Z={z∈S:z≥10,D(z)=0} and control every root, not only 20 or a finite sample. Seek a uniform rank or depth argument over Z, or justify an exhaustive reduction of its whole boundary; the height theorem for one prescribed positive target does not bound the union.
Ready to work on
02
Prove I^11({2,4,6,8})=∅ with every inverse layer fully accounted for.Suggested move: Start the nonzero even roots at persistence zero and build or bound eleven inverse steps using x≥10 at every step. Preserve all unclassified residuals and exact coverage evidence; individual layer finiteness does not certify the final layer empty.
Ready to work on
03
Close the conditional route through 20 beyond its first three known inverse layers.Suggested move: Resolve I(6144), I(16128), I(21168), and I(774144) to finish the displayed fourth layer, then pursue I^10({20})=∅. To use the 20 branch for the full zero-terminal exclusion, it suffices to force every hypothetical persistence-eleven zero-terminal smooth value into that branch. The stronger threshold-six assertion is one sufficient option, not a necessary prerequisite; its threshold-five variant has an explicit counterexample. A direct proof of I^10(Z)=∅ could avoid concentration at 20.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen conjecture

The universal decimal upper bound of eleven iterations remains open in the cited literature. Odd terminal digits have a source-reported solution. The August 2026 preprint establishes a valuation bound used for further finite analysis of nonzero even terminal digits; it does not state a complete proof for every decimal integer.

[3][4]
External progress

What the literature has established

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

  1. PreprintNyadjo Fonga reports a uniform 2-adic valuation bound for fixed multiplicities of digits 2 through 9 with arbitrary insertions of digit 1.[4]
  2. PreprintBrier, Clavier, Gutsche, and Naccache report bounds of one step for terminal digits 1, 3, 7, 9, and five steps for terminal digit 5.[3]
  3. Peer reviewedPerez and Styer report a decimal search range below 10^1500 with no persistence above eleven, alongside results in other bases. This is finite evidence.[2]
  4. Historical sourceSloane formulates bounded persistence in each base and reports no decimal example above eleven below 10^50. The original boundedness conjecture does not fix its constant to eleven.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusMultiplicative persistence conjecture
Weaker or relaxed formExistence of some uniform decimal persistence bound

Sloane's boundedness conjecture asks for an unspecified constant; the target value eleven is a stronger assertion.

[1]
Solved special caseOdd terminal digits

The 2021 result conditions on the final digit, not on whether the starting integer is odd.

[3]
Dependency or reductionValuation bounds for nonzero even terminal analysis

The 2026 preprint supplies an explicit finite-analysis ingredient. Further analysis and the zero terminal case remain outside its stated conclusion.

[4]

Formal and computational footholds

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

  • computation · not independently reproducedPerez–Styer finite persistence calculations

    The published paper describes Maple-based calculations and reports the decimal range below 10^1500. No computation was repeated here.

    [2]

Formalization opportunities

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

  • Formalization targetA proposed formal start is decimal digit product with zero and one-digit conventions, then a termination proof distinct from a uniform step bound.
  • Formalization targetFuture use of the terminal-digit results needs exact domain and valuation hypotheses; a finite algorithm also needs its final execution and completeness evidence.

Detailed research inventory

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
  • lemma3 of 93
  • equivalence1 of 91
  • computational claim2 of 92
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 statementP(n)≤11 for every n≥0, where P counts digit-product steps until one digit remains.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementA positive first product reduces the target to smooth numbersintermediate
  • retained route statementAn explicit trajectory attains eleven stepsintermediate
  • retained route statementEvery fixed positive inverse problem is finiteintermediate
  • retained route statementSeventeen fixed targets have reported complete predecessor setsintermediate
  • retained route statementThe inverse tree above 20 has a defined next frontierintermediate
  • retained route statementThe zero and nonzero-even branches have different indicesintermediate
  • 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 failureForce every smooth zero-terminal trajectory of persistence at least five through 20.reported failure
  • Useful failureInfer actual smooth trajectories from a fixed-modulus relaxation that inspects only bounded leading and trailing digits.reported failure
  • Research targetProve I^10(Z)=∅ for the entire infinite positive zero boundary Z.open
  • Research targetProve I^11({2,4,6,8})=∅ with every inverse layer fully accounted for.open
  • Research targetClose the conditional route through 20 beyond its first three known inverse layers.open
  • ComputationThe current work reports seventeen complete inverse sets, 28 certified valuation maxima, and a forward enumeration of 438,492 reduced smooth values below 10^60. That scan finds two persistence-ten values, 4996238671872 and 937638166841712, both mapping to 438939648.The finite scan does not show that only two persistence-ten smooth numbers exist. The second audit did not rerun the largest exhaustive producer workloads and created no new inverse set. The seventeen classifications require the full producer and all stated tail exclusions; a short regression, timeout, or unknown residual is not an empty inverse set. · reported unreproduced
  • Narrowed routeForce every smooth zero-terminal trajectory of persistence at least five through 20.The source gives 279988627968→2633637888→6967296→244944→4608→0. The starting value is smooth and has persistence five, so the example defeats the threshold-five statement. It does not prove the proposed threshold-six statement. A threshold-six forcing theorem remains an open candidate. A direct uniform bound above all of Z avoids needing that strengthening.
  • Narrowed routeInfer actual smooth trajectories from a fixed-modulus relaxation that inspects only bounded leading and trailing digits.The current work constructs safe inspected ends for arbitrarily large powers of two and relaxed edges witnessed by other decimal numbers with matching ends, length, and residue. Those witnesses need not be the smooth state itself, so D(s_i)=s_{i+1} can fail throughout an arbitrarily long relaxed path. Adaptive moduli, target-dependent bounds, and whole-word arguments remain available. The limited obstruction does not rule out every finite-state or modular proof.
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 I^10(Z)=∅ for the entire infinite positive zero boundary Z.

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 I^10(Z)=∅ for the entire infinite positive zero boundary Z.

Multiplicative persistence conjecture · ready to start

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Research contextPrepared context for any AI agent

Multiply a number’s decimal digits and repeat until one digit remains. The conjecture says this always takes at most eleven steps. The current work classifies seventeen particular inverse problems and proves a finite bound for predecessors of each fixed positive target, while the depth of all possible trajectories remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 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. 1
    The Persistence of a Numberoriginal source · N. J. A. Sloane · Journal of Recreational Mathematics 6, 97–98 · 1973 · accessed Sep 7, 2026
  2. 2
    Persistence: a digit problempeer reviewed result · Stephanie Perez, Robert Styer · Involve 8, 439–446 · 2015 · DOI 10.2140/involve.2015.8.439 · accessed Sep 7, 2026
  3. 3
    The Multiplicative Persistence Conjecture Is True for Odd Targetspreprint · Eric Brier, Christophe Clavier, Linda Gutsche, David Naccache · arXiv · 2021-10-08 · ARXIV 2110.04263 · accessed Sep 7, 2026
  4. 4
    The Multiplicative Persistence Conjecture: Resolving the 2-Adic Obstruction for Nonzero Even Targetspreprint · Patrick Nyadjo Fonga · arXiv · 2026-08-28 · ARXIV 2608.27802 · 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.
  • A proposed year for the exact eleven-step assertion is not assigned: the inspected indexed text of Sloane's 1973 paper states an unspecified bound, while later sources attribute the stronger form to that tradition.
  • The original PDF's second-page screenshot failed; its indexed primary text and the author's publication listing supplied the historical binding.
  • MSC 00A08 appears in Perez–Styer. The recent even-target preprint is not treated as a complete proof or a completed finite check.

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