Number theory · exponential Diophantine equations · arithmetic geometry

Pillai's Conjecture

Collaboration beta

The v8 source keeps Pillai's fixed-gap conjecture open. It replaces the stale affine-ray/Kummer frontier with a maximal-exponent cubic reduction, all-fixed-parameter denominator and coefficient-primitivity theorems, exact balanced and Cardano saturation, and a two-colour norm-one S-unit interface of height 2 log Y+o(log Y). The terminal arithmetic theorem is missing. Five live routes, six sufficient blockers, seven work orders, and ten compact no-revisit findings define the current source-reported frontier. The supporting materials have not been independently checked and are not independent evidence.

x^{m} - y^{n} = k \Rightarrow finitely many (x, y, m, n) for fixed k \neq 0

Known results and sources

Pillai's conjecture asks whether a fixed nonzero gap can occur between pure perfect powers only finitely many times.

Research problem

Exact mathematical statement

Fix an integer $k\ne 0$ . Pillai's conjecture asks whether the equation

x^m-y^n=k

has only finitely many tuples of integers with $x,y\ge 2$ and $m,n\ge 2$ . Swapping the two perfect powers replaces $k$ by $-k$ , so the two signs are treated symmetrically. The selected v8 source develops a maximal-exponent cubic reduction and a two-colour norm-one S-unit program; the terminal arithmetic theorem remains open.

Problem infographic

Problem at a glance

Historical v6 route diagram showing affine and Kummer structures, retained only for context; the current v8 cubic and two-colour frontier appears in the research map below. — Historical v6 affine/Kummer route diagram retained for context. The current v8 frontier is the maximal-exponent cubic and two-colour program shown in the research map below; its terminal arithmetic theorem remains open.

Current mathematical picture

Where work on Pillai's Conjecture stands

Open conjecture

Leading routeR1 two-colour primitive-divisor route

Work directly with 1-𝔲=ηZ^MW^N and seek a theorem specialized to a norm-one element in a pure cubic field, such as a nondegenerate conjugate recurrence or primitive positive valuation incompatible with both exponent colours.

Route status · Active route

Main reductionT-CANONICAL maximal-exponent reduction

Along a hypothetical infinite fixed-gap counterfamily with fixed k at least two, each numerical perfect power has a unique maximal-exponent representation; after discarding bounded values and using the source's fixed-prime exclusion, the exponents and bases are pairwise coprime in the stated senses.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeB1 primitive valuation outside the kernel

Prove that for all sufficiently large coprime M,N the special norm-one element has a prime ideal outside S with a positive valuation of 1-𝔲 divisible by neither M nor N.

Task status · Ready to work on

Latest mathematical updatev8 replaces the stale affine frontier with the cubic two-colour frontier

The v8 source reports a maximal-exponent cubic reduction, fixed-parameter denominator and primitivity results, saturation of several older routes, and a two-colour norm-one interface while leaving the terminal theorem open.

Retained source record

Work mapped so far

Pillai's Conjecture in numbers

3.4kretained lines of mathematical investigation1,045 in the current working snapshot

Argument development: 2,827 · 84%
Explored or eliminated routes: 107 · 3%
Computational analysis: 38 · 1%
Open obligations: 194 · 6%
Definitions and setup: 209 · 6%

24inventoried working statements9routes investigated13open questions13contribution-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 byS. S. PillaiStatus checkedAug 25, 2026

Source recordedAug 27, 2026

Recommended next task

B1 primitive valuation outside the kernel

Prove that for all sufficiently large coprime M,N the special norm-one element has a prime ideal outside S with a positive valuation of 1-𝔲 divisible by neither M nor N.

Suggested move: Seek a primitive-divisor or valuation-one theorem specialized to the exact two-colour pure-cubic norm-one family.

What would count as progress

Produce a prime ideal 𝔭 outside S
Prove v_𝔭(1-𝔲)>0
Prove neither M nor N divides that valuation

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

28 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 routeR1 two-colour primitive-divisor route

Route status · Active route

Active routeR2 special Cardano-height route

Exploit both perfect-power factor identities on the Mordell curve; a useful theorem must retain them and obtain polynomial dependence on rad(g) or AB.

Route status · Active route

Active routeR3 additive high-power residue route

Use the exact special residue and both high-power twists to produce a forbidden short multiplier; the companion cubic congruences themselves add no information.

Route status · Active route

Active routeR4 two-exponent modular route

Seek a Frey representation or modular obstruction from the fixed-gap equation, pure cubic norm equation, or two-colour identity, and audit conductor growth before large computations.

Route status · Active route

Active routeR5 moving-field ideal route

Factor (ξ) and (1-𝔲) in the pure cubic field while keeping explicit discriminant, regulator, class-group, and support dependence; do not replace ideal powers by element powers without class-group control.

Route status · Active route

Explored alternatives

Other routes

4 recorded

Useful but insufficientInfer Frey-Jacobian endomorphisms from the affine mod-2 branch action

The second audit withdraws that inference and retains only the genus, discriminant, nodal-thickness, and residual-conductor statement. Use the Frey-Jacobian conductor theorem only, pending a separately proved decomposition or residual-image theorem.

Route status · Useful but insufficient

Useful but insufficientContain the full Kummer field in one rank-two Dickson torsion field

The current work's Sylow-rank obstruction rules out that direct containment; only a boundary/Prym split across two reflection factors remains supported. Construct two generic normal functions whose complementary boundary classes recover the two Kummer halves and then prove an arithmetic residual-image restriction.

Route status · Useful but insufficient

Route held in reserveN2 crossed-quotient reciprocity route

The source separately requires the original and swapped fixed-field ray quotients to be compared in one normal closure or through their Artin conductors, seeking a crossed-exponent reciprocity relation that makes the two affine constituents incompatible.

Route status · Route held in reserve

Browse 1 more explored route

Route held in reserveN7 modular fallback route

The source separately requires the degree-prime transvection from DEG-BLOW-01 to enter the residual-image analysis, followed by classification of every reducible case with its local characters and fixed conductor support before automorphy or level lowering.

Route status · Route held in reserve

Route statements and reductions

Statements the next route can inspect and build on

Route statementT-ANTI-SHORT arbitrary determinant anti-short theorem

If a companion column has determinant Δ=Y^{o(1)} and 0<|N_c|≤Y, the source proves |N_c|=Y^{1-o(1)}.

Source-reported route statement · dependencies incomplete

Route statementT-COMPANION-PARAM exact companion parametrization

For the source's integer background Xn−Ym=1 and any integer companion column (M_c,N_c)∈ℤ², set Δ=XN_c−YM_c and h=M_cn−N_cm; then M_c=mΔ+Xh and N_c=nΔ+Yh, with the stated exact congruence and valuation consequences.

Source-reported route statement

Route statementT-CONGRUENCE-REDUNDANCY companion cubic congruences add no phase

After substituting the exact companion parametrization, the two companion cubic congruences reduce to the adjacent congruences multiplied by Δ^3, so they provide no independent CRT phase.

Source-reported route statement

Route statementT-CARDANO exact toric Cardano ring

For H=𝔄^2-k𝔅 and the source's Cardano factors 𝕌,𝕍, the exact identities H=gXY, 𝕌=gγY^3, 𝕍=gαX^3, 𝕌𝕍=H^3, and 𝕍-𝕌=kg hold, yielding the stated toric coefficient ring after inverting fixed 2k.

Source-reported route statement

Route statementT-CARDANO-SATURATION mixed no-excess theorem

For a nonzero homogeneous Cardano section F of degree D with v_X(F)≥r and v_Y(F)≥s, the source proves first local weight ν(F)≤D-r-s; hence fixed-degree universal interpolation cannot save a fixed power beyond its visible divisor.

Source-reported route statement · dependencies incomplete

Route statementT-TWO-COLOR-SUNIT exact norm-one equation

With τ^3=α^2γ, K=Q(τ), ξ=αX−τY, 𝔲=ξ^3/(α^2k), and η=3τξ/(kα^{j_M+1}γ^{j_N}), the source proves N(𝔲)=1 and 1−𝔲=ηZ^MW^N. For every prime ideal outside S, where S consists of the prime ideals above 3kαγ, η has valuation zero and every positive valuation of 1−𝔲 is an M-multiple or an N-multiple according to two disjoint rational supports: gcd(A,B)=1, supp(Z)=supp(A), and supp(W)=supp(B).

Source-reported route statement · dependencies incomplete

Route statementT-SUNIT-HEIGHT exact asymptotic height

For the special norm-one element 𝔲, the source proves h(𝔲)=2 log Y+o(log Y), records sublinear field discriminant and logarithmic radical, and warns that generic S-unit estimates may lose force through moving regulator, class-group, or coefficient-height dependence.

Source-reported route statement · dependencies incomplete

Route statementMissing terminal arithmetic theorem

No terminal contradiction is proved. Completion requires a new arithmetic theorem using both unbounded exponent colours simultaneously—such as a primitive positive valuation, support-sensitive height collapse, special Cardano height bound, forbidden additive rotation, or controlled-level two-exponent modular obstruction—with sufficiently uniform dependence on moving data.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

13 featured tasks

B1 primitive valuation outside the kernel

Prove that for all sufficiently large coprime M,N the special norm-one element has a prime ideal outside S with a positive valuation of 1-𝔲 divisible by neither M nor N.

Suggested move: Seek a primitive-divisor or valuation-one theorem specialized to the exact two-colour pure-cubic norm-one family.

Ready to work on

W1 principalization-resistant two-colour ideal analysis

Study ideal quotients that cancel (η) outside the kernel without choosing generators and test whether class-group powers or conjugate ideal ratios yield a recurrence with incompatible new-prime multiplicities.

Suggested move: Track discriminant, class-number, and regulator dependence before principalizing.

Ready to work on

B6 terminal uniformity boundary

Translate every proposed moving-field, primitive-divisor, modular, or height theorem into polynomially effective dependence on all data that are only subpower in Y.

Suggested move: Record dependence on regulator, class number, coefficient height, and support size before treating an imported theorem as live.

Ready to work on

B3 polynomial special-Cardano height

Prove a bound Y≪_k rad(g)^C for the Cardano points satisfying both high-power factor identities.

Suggested move: Use both perfect-power Cardano factors; do not discard them through a generic Mordell estimate.

Ready to work on

B4 additive rotation theorem

Produce a fixed ε>0 and a nonzero subpower multiplier Δ such that ||nΔ/Y||≤Y^{-ε} using the special residue and high-power twists.

Suggested move: Exploit n≡k^{-1}αX^2 mod Y together with both high-power twists; ordinary Dirichlet approximation is insufficient.

Ready to work on

B5 two-exponent modular/Frey route

Construct a Frey object seeing both moving exponents with level controlled by kαγ and incompatible local behavior, rather than conductor growth through all primes of AB.

Suggested move: Audit the candidate object's conductor growth and local minimality before any large computation.

Ready to work on

W3 special Mordell/Belyi factor theorem

Use the simultaneous Mordell equation and both perfect-power factor identities, or the equivalent Belyi identity, to seek a polynomial height bound.

Suggested move: Audit factor gcds and ideal-class control before invoking logarithmic forms.

Ready to work on

W4 additive residue search with exact high-power structure

Seek a multiplicative-subgroup, exponential-sum, or sparse-orbit result for the structured residue n/Y modulo the moving powerful modulus that produces a forbidden subpower multiplier.

Suggested move: Work with the exact high-power congruence; average equidistribution is insufficient.

Ready to work on

W2 conjugate-difference recurrences

Study ξ_i/ξ_j and 1-(ξ_i/ξ_j)^3 to determine whether they form a nondegenerate binary or ternary recurrence after kernel units are removed.

Suggested move: First prove or refute genuine nondegeneracy, then seek valuation incompatibility with both exponent colours.

Ready to work on

W6 uniform primitive-divisor literature scan

Search primary literature for primitive-divisor and valuation-one theorems over number fields and record exact dependence on the moving field, discriminant, regulator, initial term, and index.

Suggested move: Reject fixed-sequence uniformity that does not remain uniform when both the sequence and field move.

Ready to work on

W5 two-exponent Frey feasibility audit

Identify a curve or representation seeing both M and N, determine whether its conductor avoids all primes of AB, test level lowering at both supports, and check incompatibility of the two local conditions.

Suggested move: Abort the modular route early if conductor growth loses the exponent information.

Ready to work on

B2 support-sensitive height theorem

In the special degree-three family, prove that sublinear field discriminant, sublinear logarithmic radical, and two disjoint positive multiplicity colours force h(𝔲)=o(log Y).

Suggested move: Control the dependence on η, the moving regulator, and the class group instead of applying a generic S-unit bound.

Ready to work on

W7 independent reconstruction of novel interfaces

Independently reconstruct T-PAIR-PRIMITIVITY and T-CARDANO-SATURATION before relying on either in a terminal proof.

Suggested move: Recheck rational-section denominator clearing, fixed-shear signs, toric monomial filtration, and cusp injectivity independently.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 25, 2026

Current statusOpen conjecture

The cited 2015 literature describes Pillai's fixed-gap finiteness conjecture as widely open. It provides context and restricted methods, not a proof for every fixed nonzero gap with both bases and exponents variable.

[1][2]

External progress

What the literature has established

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

2021
PreprintBennett and Siksek develop explicit Diophantine, modular, and computational methods for restricted prime-power-gap equations and solve specified families.[3]
2015
Peer reviewedDietmann and Elsholtz describe the fixed-gap assertion for pure powers as widely open while using it as context for a separate Hilbert-cube problem.[2]
2000
Peer reviewedShorey records the generalized equation a x^m - b y^n = k and Pillai's conjecture that it has finitely many solutions under the stated exponent conditions.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusPillai's conjecture

Stronger or generalized formgeneralized Pillai equation a x^m-b y^n=k

Shorey's displayed formulation permits fixed positive coefficients a and b, so the pure-power difference x^m-y^n=k is its coefficient-one specialization.

[1]

Related problemHilbert cubes in pure powers

Hilbert cubes in the set of pure powers are related through repeated bounded gaps, but their finite-dimensionality does not prove the full fixed-gap conjecture.

[2]

Solved special caserestricted prime-power-gap equations

Restricted equations with a square on one side and a difference supported on prescribed prime powers can be explicitly determined in many cases; these are narrower families than the unrestricted fixed-k statement.

[3]

Formal and computational footholds

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

computation · not independently reproducedRestricted prime-power-gap calculations
The preprint reports explicit determination of specified families using Diophantine approximation, modular methods, and equation-solving computations; this metadata lane did not reproduce them.
[3]

Formalization opportunities

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

Formalization targetFormalize the exact v8 companion parametrization, Cardano saturation, two-colour norm-one S-unit identity, and the remaining support-sensitive terminal arithmetic interface; these remain source-reported until independently checked.

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.

v8 replaces the stale affine frontier with the cubic two-colour frontierThe v8 source reports a maximal-exponent cubic reduction, fixed-parameter denominator and primitivity results, saturation of several older routes, and a two-colour norm-one interface while leaving the terminal theorem open.

Changed the research frontierLater mathematical revision

Cumulative v8 source ingested; not a claim of mathematical occurrence time

v6 refines the affine Kummer and one-prime frontierThe cumulative v6 source reports exact prime-core, affine Kummer, local-divisibility, and one-prime packet advances while keeping seven global interfaces open.

Changed the research frontierLater mathematical revision

Cumulative v6 source ingested; not a claim of mathematical occurrence time

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

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 inventory covers all currently cataloged mathematical statements in the research notes.

22 standing statements2 proposed statements13 open questions1 conditional results

Statements by mathematical role24 mapped statements

negative result3 of 243
theorem candidate2 of 242
reduction3 of 243
lemma16 of 2416

Complete mathematical inventory1 mathematical clusters

Pillai v8 recorded theorem and frontier mapThe v8 active projection records the maximal-exponent and cubic reductions, all fixed-parameter denominator and primitivity results, exact balanced and Cardano saturation, the two-colour norm-one interface, the missing terminal theorem, six blockers, five live routes, seven work orders, and ten compact no-revisit findings. All statuses remain source-reported and provisional.70 displayed rows · 5 routes included

retained route statementPillai fixed-gap finiteness target
retained route statementT-CANONICAL maximal-exponent reductionintermediate
retained route statementEXT-BU bounded-support exclusionintermediate
retained route statementEXT-W2 Baker product windowintermediate
retained route statementT-KERNEL fixed-r kernel growthintermediate
retained route statementT-CUBIC determinant-one transformed cubicintermediate
retained route statementT-TWIST exact high-power twistsintermediate
retained route statementT-ROOT ordered-root modelintermediate
retained route statementT-TRIANGLE ordinary triangular jet isomorphismsintermediate
retained route statementT-EXCEPTIONAL characteristic-two exceptional ranksintermediate
retained route statementT-DEN-ALL all-fixed-parameter denominator theoremintermediate
retained route statementT-ANTI-SHORT arbitrary determinant anti-short theoremintermediate
retained route statementT-PAIR-PRIMITIVITY fixed-R coefficient primitivityintermediate
retained route statementT-COMPANION-PARAM exact companion parametrizationintermediate
retained route statementT-CONGRUENCE-REDUNDANCY companion cubic congruences add no phaseintermediate
retained route statementT-QNORM differential sextic quadratic normintermediate
retained route statementT-BALANCED-SATURATION critical lattice scalesintermediate
retained route statementT-BALANCED-HEIGHT exact critical cubic heightintermediate
retained route statementT-CARDANO exact toric Cardano ringintermediate
retained route statementT-CARDANO-SATURATION mixed no-excess theoremintermediate
retained route statementT-TWO-COLOR-SUNIT exact norm-one equationintermediate
retained route statementT-SUNIT-HEIGHT exact asymptotic heightintermediate
retained route statementMissing terminal arithmetic theoremconditional
Recorded relationshipThe main chain applies the fixed-r kernel bookkeeping after the canonical counterfamily reduction; the theorem-ledger row records EXT-BEG plus primewise bookkeeping as the kernel theorem's controlling inputs.supports · reported by source
Recorded relationshipThe ordered-root model is supported by the exact factorization of the transformed cubic.supports · reported by source
Recorded relationshipThe ordinary triangular jet isomorphisms use the restriction recursion in the ordered-root model.supports · reported by source
Recorded relationshipThe exceptional-rank branch uses the transfer and associated matrices within the same ordered-root machinery.supports · reported by source
Recorded relationshipThe theorem ledger names the preceding ordered-root, triangular, and exceptional results as the denominator theorem's prerequisites.supports · reported by source
Recorded relationshipThe theorem ledger gives the jet-theorem machinery plus the differential-sextic start as the anti-short theorem's prerequisites; T-DEN-ALL is not asserted as a direct premise here.supports · reported by source
Recorded relationshipThe theorem ledger identifies triangular isomorphism and the denominator theorem as the primitivity theorem's main inputs.supports · reported by source
Recorded relationshipThe exact companion parametrization follows from the determinant-one setup of the transformed cubic.supports · reported by source
Recorded relationshipThe theorem ledger states that the no-new-phase conclusion depends on the exact companion parametrization.supports · reported by source
Recorded relationshipThe theorem ledger identifies pure cubic algebra as the quadratic-norm theorem's input.supports · reported by source
Recorded relationshipThe theorem ledger identifies the sextic lower bound as the balanced-saturation theorem's input.supports · reported by source
Recorded relationshipThe theorem ledger gives saturation plus exact expansion as the critical balanced-height inputs.supports · reported by source
Recorded relationshipThe theorem ledger identifies the transformed cubic's Hessian/Cardano identities as the exact Cardano-ring input.supports · reported by source
Recorded relationshipThe theorem ledger identifies toric filtration and cusp injectivity in the exact Cardano ring as the mixed no-excess inputs.supports · reported by source
Recorded relationshipThe theorem ledger names the norm identity, exact high-power twists, and support containment as the two-colour S-unit inputs. T-CUBIC supplies the exact cubic signature behind the norm identity, T-TWIST supplies the two high-power factors, and T-KERNEL supplies the cubic-kernel setting.supports · reported by source
Recorded relationshipThe theorem ledger identifies the conjugate and finite-place sizes of the special two-colour element as the height theorem's inputs.supports · reported by source
Recorded relationshipThe exact main-chain endpoint records that these saturated structural results lead only to a missing terminal arithmetic theorem.supports · reported by source
Recorded relationshipThe source states that a suitable terminal arithmetic theorem would contradict the hypothetical canonical counterfamily and finish the fixed-gap target.supports · reported by source
Useful failureCoupled cubic-root CRT from companion congruencesreported failure
Useful failureLarge coefficient content and powerful Pell descentreported failure
Useful failureRecursive cubic content amplificationreported failure
Useful failureChoose a better balanced lattice basisreported failure
Useful failureFixed-degree Cardano interpolationreported failure
Useful failureGeneric degree-growing coefficient-height interpolationreported failure
Useful failureGeneric Mordell or Thue boundsreported failure
Useful failureOne-sided powerful-modulus heuristicreported failure
Useful failurePure quadratic-norm rigidity at -kreported failure
Useful failureFinite endpoint interpolation at selected heightsreported failure
Research targetB1 primitive valuation outside the kernelopen
Research targetB2 support-sensitive height theoremopen
Research targetB3 polynomial special-Cardano heightopen
Research targetB4 additive rotation theoremopen
Research targetB5 two-exponent modular/Frey routeopen
Research targetB6 terminal uniformity boundaryopen
Research targetW1 principalization-resistant two-colour ideal analysisopen
Research targetW2 conjugate-difference recurrencesopen
Research targetW3 special Mordell/Belyi factor theoremopen
Research targetW4 additive residue search with exact high-power structureopen
Research targetW5 two-exponent Frey feasibility auditopen
Research targetW6 uniform primitive-divisor literature scanopen
Research targetW7 independent reconstruction of novel interfacesopen
ComputationThe governing source reports an exact finite Cardano saturation enumeration through degree 22.The source reports that 2,116 nonzero valuation subspaces were checked; the computation is evidence for, not a premise of, T-CARDANO-SATURATION. · reported unreproduced
Active routeR1 two-colour primitive-divisor routeWork directly with 1-𝔲=ηZ^MW^N and seek a theorem specialized to a norm-one element in a pure cubic field, such as a nondegenerate conjugate recurrence or primitive positive valuation incompatible with both exponent colours.
Active routeR2 special Cardano-height routeExploit both perfect-power factor identities on the Mordell curve; a useful theorem must retain them and obtain polynomial dependence on rad(g) or AB.
Active routeR3 additive high-power residue routeUse the exact special residue and both high-power twists to produce a forbidden short multiplier; the companion cubic congruences themselves add no information.
Active routeR4 two-exponent modular routeSeek a Frey representation or modular obstruction from the fixed-gap equation, pure cubic norm equation, or two-colour identity, and audit conductor growth before large computations.
Active routeR5 moving-field ideal routeFactor (ξ) and (1-𝔲) in the pure cubic field while keeping explicit discriminant, regulator, class-group, and support dependence; do not replace ideal powers by element powers without class-group control.

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 that for all sufficiently large coprime M,N the special norm-one element has a prime ideal outside S with a positive valuation of 1-𝔲 divisible by neither M nor N.

The current research map records this as an open mathematical step.

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.

Produce a prime ideal 𝔭 outside S
Prove v_𝔭(1-𝔲)>0
Prove neither M nor N divides that valuation

Continue the mathematics

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Prepared starting pointB1 primitive valuation outside the kernel

Pillai's Conjecture · ready to start

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Research contextPrepared context for any AI agent

For a fixed nonzero gap, can only finitely many pairs of perfect powers differ by exactly that amount?

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Current routes and known obstacles
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Sources and references3 cited works · next context review by Nov 25, 2026

The mathematical context was checked on Aug 25, 2026. Status can be refreshed sooner after a material result or claim.

1
Some conjectures in the theory of exponential Diophantine equationssurvey or monograph · T. N. Shorey · Publicationes Mathematicae Debrecen · 2000 · accessed Aug 25, 2026
2
Hilbert cubes in arithmetic setspeer reviewed result · Rainer Dietmann, Christian Elsholtz · Revista Matemática Iberoamericana · 2015 · DOI 10.4171/RMI/877 · accessed Aug 25, 2026
3
Differences between perfect powers: prime power gapspreprint · Michael A. Bennett, Samir Siksek · arXiv · 2021-10-11 · ARXIV 2110.05553 · accessed Aug 25, 2026

Important qualifications

The bounded primary-source search establishes representative formulations and restricted advances, not an exhaustive history of exponential Diophantine equations.
The Bennett–Siksek work treats specified prime-power-gap families and does not settle the unrestricted fixed-gap conjecture.
The conjecture's literature status is supported by the cited external sources, not by the submitted materials or their reported calculations.
No statement-aligned formalization of the full fixed-gap finiteness conjecture was established by this scoped search.

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