[#P8] Fermat-Catalan conjecture

Problem. There are only finitely many primitive solutions \((a,b,c,m,n,k)\in\mathbb{Z}_{>0}^6\) to \(a^m+b^n=c^k\) with \(m,n,k\ge2\) and \(1/m+1/n+1/k<1\).

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Definitions and notation

1Context

The conjecture unifies several generalized Fermat equations through a finiteness claim.

2Problem setup

Definition 1 (A solution). A solution is primitive when a, b, and c have greatest common divisor 1.

Remark 1. The reciprocal-sum condition selects the genuinely hyperbolic exponent triples and excludes infinite elementary families.

3What counts as a solution

Prove that the set of primitive solutions satisfying the exponent condition is finite, or construct and verify infinitely many distinct primitive solutions.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The linked 2024 work formulates product variants and treats the classical primitive-solution finiteness statement as conjectural. Darmon and Granville prove finiteness when the exponent triple is fixed, while the union over varying exponents remains open. Exact unresolved remainder: Prove that only finitely many primitive solutions satisfy the reciprocal-exponent condition, or construct infinitely many distinct primitive solutions.^[1][2]

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The linked 2024 work formulates product variants and treats the classical primitive-solution finiteness statement as conjectural. Darmon and Granville prove finiteness when the exponent triple is fixed, while the union over varying exponents remains open. Exact unresolved remainder: Prove that only finitely many primitive solutions satisfy the reciprocal-exponent condition, or construct infinitely many distinct primitive solutions.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The linked 2024 work formulates product variants and treats the classical primitive-solution finiteness statement as conjectural. Darmon and Granville prove finiteness when the exponent triple is fixed, while the union over varying exponents remains open. Exact unresolved remainder: Prove that only finitely many primitive solutions satisfy the reciprocal-exponent condition, or construct infinitely many distinct primitive solutions.^[1][2] Relevance to this problem For Fermat-Catalan conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The linked 2024 work formulates product variants and treats the classical primitive-solution finiteness statement as conjectural. Darmon and Granville prove finiteness when the exponent triple is fixed, while the union over varying exponents remains open. Exact unresolved remainder: Prove that only finitely many primitive solutions satisfy the reciprocal-exponent condition, or construct infinitely many distinct primitive solutions. Open full record →
Prove that the set of primitive solutions satisfying the exponent condition is finite, or construct and verify infinitely many distinct primitive solutions.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that the set of primitive solutions satisfying the exponent condition is finite, or construct and verify infinitely many distinct primitive solutions.^[1] Relevance to this problem For Fermat-Catalan conjecture, record fermat-catalan-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that the set of primitive solutions satisfying the exponent condition is finite, or construct and verify infinitely many distinct primitive solutions. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that the set of primitive solutions satisfying the exponent condition is finite, or construct and verify infinitely many distinct primitive solutions. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2024 paper treats the Fermat-Catalan statement as an unresolved conjecture. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Fermat's last theorem and Catalan's theorem settle important boundary cases but do not imply this finiteness statement.

Recorded example 1. The identity 2^5 + 7^2 = 3^4 is a primitive solution, and 1/5 + 1/2 + 1/4 is less than 1.

Computational notes

A bounded enumeration can enlarge the list of known solutions without proving that the complete list is finite.

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Fermat-Catalan conjecture.” TheoremDB. P8. Problem statement; statement text SHA-256 680b1dfa89f2ba6ad872fc0d57ad2bc13978b011aef5143716e14e1e4d5edc48. https://theoremdb.org/statement/?ref=P8

BibTeX

@misc{theoremdb-problem-680b1dfa89f2ba6ad872fc0d57ad2bc13978b011aef5143716e14e1e4d5edc48,
  title = {{Fermat-Catalan conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 680b1dfa89f2ba6ad872fc0d57ad2bc13978b011aef5143716e14e1e4d5edc48},
  url = {https://theoremdb.org/statement/?ref=P8}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Adam S. Sikora, “Fermat-Catalan and Tijdeman-Zagier conjectures for products”. arXiv:2410.21552 (2024). Adam S. Sikora, arXiv:2410.21552, introduction and motivating conjectures. ↗preprint · primary source · arXiv:2410.21552, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2024 paper treats the Fermat-Catalan statement as an unresolved conjecture. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at introduction, motivating conjectures, and formulation of product variants.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides a current statement of the classical boundary and formulates neighboring product conjectures without claiming their proof.Source named by the research packet.
Henri Darmon and Andrew Granville, “On the Equations z m = F ( x, y ) and Ax p + By q = Cz r”. Bulletin of the London Mathematical Society 27(6) (1995), 513-543. DOI 10.1112/blms/27.6.513. finiteness theorem for generalized Fermat equations with a fixed exponent triple. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the checked fixed-exponent finiteness result while leaving the union over varying exponents outside its conclusion.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P8] Fermat-Catalan conjecture

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