[#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 \(1/m+1/n+1/k<1\).
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.
Definition 2 (The reciprocal-sum condition selects the genuinely hyperbolic exponent triples and excludes infinite elementary families). The reciprocal-sum condition selects the genuinely hyperbolic exponent triples and excludes infinite elementary families.
Remark 1. The conjecture unifies several generalized Fermat equations through a finiteness claim.
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
Current 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]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
How to cite
TheoremDB contributors, “Fermat-Catalan conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/fermat-catalan-conjectureThis page as plain text: fermat-catalan-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.