Generalized Fermat Equation $a^3+b^3=c^n$

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq 3$, and let $(a,b,c)\in{\mathbb{Z}}^3$. Under the conventional interpretation, call the triple primitive if $\gcd(|a|,|b|,|c|)=1$, and nontrivial if $abc\neq 0$. These definitions are not specified in the supplied record and require confirmation against the cited source. No positivity or ordering restriction is supplied; negative integer coordinates are therefore included in this candidate formulation.

With these explicit interpretations, the supplied conjecture is \[ \forall n\in{\mathbb{Z}},\quad n\geq 3\ \Longrightarrow\ \nexists(a,b,c)\in{\mathbb{Z}}^3:\quad \begin{gathered} a^3+b^3=c^n,\\ abc\neq 0,\qquad \gcd(|a|,|b|,|c|)=1. \end{gathered} \] The intended meanings of primitive and nontrivial remain unresolved source qualifications. The supplied record labels the problem partially solved and its statement corrected_verified; these are supplied curation labels, not independent mathematical verification.

Short English statement

The supplied conjecture asks whether, for every integer exponent $n\geq 3$, the equation $a^3+b^3=c^n$ has no primitive integer solution with all three coordinates nonzero. The meanings of primitive and nontrivial are conventional interpretations requiring source confirmation. This is a statement-only draft requiring human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30001295$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied source citation: Explicit Methods in Number Theory (2010), Oberwolfach Reports, pp. 1843--1920, DOI 10.4171/owr/2009/33; https://doi.org/10.4171/owr/2009/33. The record attributes the collection to workshop open-problem discussions, supplies proposed year $2009$, and names no individual proposer. The supplied report title year and proposed year have not been reconciled.
• Supplied literature reference, labeled "Chen--Siksek on sums of two cubes": https://doi.org/10.1016/j.jalgebra.2009.03.010. No fuller bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that Chen and Siksek prove the conjecture for density-one sets of exponents and many explicit cases, but not for every $n\geq 3$. The listed sources checked are the OWR report and the Chen--Siksek reference above. This is supplied dated curation derived from the OWR collection, not independently verified here; the record expressly requests independent verification of time-sensitive status and cited claims.