# P8: Fermat-Catalan conjecture - ID: `P8` - Reference: `fermat-catalan-conjecture` - Page: https://theoremdb.org/statements/P8 - Record maturity: Reviewed problem with recorded work ## 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 \(1/m+1/n+1/k<1\). ### Context The conjecture unifies several generalized Fermat equations through a finiteness claim. ### Problem setup - **Definition (A solution).** A solution is primitive when a, b, and c have greatest common divisor 1. - **Definition (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.** The conjecture unifies several generalized Fermat equations through a finiteness claim. ### What 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. ## Status 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](#reference-1) [2](#reference-2) ## Work ### Evidence for the current status **Claim 1 (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. 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. ### Background and intake notes - Original intake status: The cited 2024 paper treats the Fermat-Catalan statement as an unresolved conjecture. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review. - The formulation and status were checked against the cited paper on 2026-07-22. - Fermat's last theorem and Catalan's theorem settle important boundary cases but do not imply this finiteness statement. - Recorded example: The identity 2^5 + 7^2 = 3^4 is a primitive solution, and 1/5 + 1/2 + 1/4 is less than 1. ### Open directions - **Route 1** (reported): Prove that the set of primitive solutions satisfying the exponent condition is finite, or construct and verify infinitely many distinct primitive solutions. [1](#reference-1) ### Computational notes - A bounded enumeration can enlarge the list of known solutions without proving that the complete list is finite. ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `fermat-catalan-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. 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 https://arxiv.org/abs/2410.21552 - Also cited at introduction, motivating conjectures, and formulation of product variants - Also cited at Editorial research route recorded 2026-07-31 - preprint; primary source; arXiv:2410.21552, checked 2026-07-31; checked 2026-07-31 - Source 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. - 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. 2. 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 https://doi.org/10.1112/blms/27.6.513 - journal_article; primary source; checked 2026-08-01 - Source use: original_summary - Proves the checked fixed-exponent finiteness result while leaving the union over varying exponents outside its conclusion.