\section{Introduction}\label{sec:intro}

\subsection{The equation}
A solution of $x^3+y^3=z^n$ in \emph{coprime nonzero integers} is a triple $(a,b,c)$ of nonzero integers with $\gcd(a,b,c)=1$ and $a^3+b^3=c^n$.  For $n=2$ there are such solutions, for example $1^3+2^3=3^2$.  For $n\ge 3$ none is known, and none is expected.  The case $n=3$ is Euler's.  For prime exponents $p\ge 17$ the equation has been attacked by the modular method since Darmon and Granville \cite[Proposition~4.4]{DarmonGranville95}, who attached to a solution a Frey curve and, assuming the Shimura--Taniyama conjecture, excluded solutions with $c$ even.  Kraus~\cite{Kraus98} wrote this curve as
\[
E_{a,b}\colon\ Y^2=X^3+3abX+b^3-a^3
\]
and proved that there is no solution for $17\le p<10^4$, and Dahmen \cite[\S3.3.2]{Dahmen08} extended his argument to $p=5,7,11,13$.  Chen and Siksek proved that there is no solution for $3\le n\le 10^9$ \cite[Theorem~2]{ChenSiksek09}, and none for the prime exponents in a set of Dirichlet density $28219/44928$ \cite[Theorem~1 and \S10]{ChenSiksek09}.  Freitas~\cite[Theorem~3]{Freitas16} proved that there is no solution for any prime $p\equiv 2\pmod 3$ with $p\ge 17$.  Bennett and Das \cite{BennettDas25} proved that for every solution $(a,b,c)$ with $n\ge3$ the greatest prime divisor of $a-b$ exceeds $10^{12}$.  What these results leave open is the set of primes $p\equiv 1\pmod 3$ above $10^9$ outside the congruence classes of Chen and Siksek.

The obstruction is the Catalan solution $1^3+2^3=3^2$.  For every prime $p\ge 17$ and every solution of $x^3+y^3=z^p$, the mod~$p$ representation of Kraus's Frey curve is isomorphic to that of the elliptic curve $W=E_{2,1}\colon Y^2=X^3+6X-7$ of conductor $72$ (Cremona label 72a1), the Frey curve of $2^3+1^3=3^2$, or to that of its quadratic twist by $-1$; in Kraus's normalization of the solution, a choice of the order of $a,b$ and of the sign of $(a,b,c)$, it is that of $W$ \cite[Th\'eor\`eme~6.1 and Proposition~6.3]{Kraus98}.  We refer to this isomorphism as \emph{Kraus's congruence}.  So the modular method, with this Frey curve, cannot conclude by comparing Hecke eigenvalues: $W$ is a genuine elliptic curve whose mod~$p$ representation imitates that of a solution.  Freitas separated the two for $p\equiv 2\pmod 3$ by comparing symplectic types, for the twists by $-3$ of the two curves: a criterion he proved at~$2$ shows that the isomorphism is symplectic, and the criterion of Kraus and Oesterl\'e at the multiplicative prime~$3$ shows that it is symplectic only if $-3$ is a square modulo~$p$ \cite[\S2]{Freitas16}.  When $p\equiv1\pmod 3$ the two criteria agree.  As Billerey, Chen, Dieulefait and Freitas put it \cite[Remark~1.7]{BCDF-I}, the Catalan solution $2^3+1^3=3^2$ ``currently prevents the modular method from proving'' such results for $x^3+y^3=z^p$.

\subsection{The results}
\begin{maintheorem}\label{thm:main-uncond}
For every integer $n\ge3$, the equation $x^3+y^3=z^n$ has no solution in coprime nonzero integers.
\end{maintheorem}

For $3\le n\le10^9$ this is due to Chen and Siksek \cite[Theorem~2]{ChenSiksek09}, and for the prime exponents $p\equiv2\pmod3$ with $p\ge17$ to Freitas \cite[Theorem~3]{Freitas16}.  We prove the case of every prime exponent $p\equiv1\pmod 3$ with $p\ge17$ (\cref{thm:three-main}), which contains the cases left open, and \cref{thm:main-uncond} is \cref{cor:three-all}.  \Cref{sec:intro-inputs} lists the published results on which the proof rests.

The proof replaces Kraus's Frey curve by a Frey motive of rank three over $K=\Q(\sqrt{-3})$.  At a solution the parameter of the motive is $3$-adically so close to a point of maximally unipotent monodromy that inertia at $\sqrt{-3}$ acts unipotently on the fiber, whereas the parameter of $1^3+2^3=3^2$ is farther away.  By a theorem of Calegari, Emerton and Gee, the mod~$p$ representation of the fiber, twisted by a cubic character, is the reduction of a member of a compatible system of minimal ramification.  We show that the member of that system at a prime above $3$ is irreducible and ordinary at $\sqrt{-3}$.  This member is unramified outside $\sqrt{-3}$, and its ramification there is so small that, by Odlyzko's discriminant bounds and finite group theory, every reduction of it has trivial semisimplification.  As the Galois group of the maximal pro-$3$ extension of $K$ unramified outside $\sqrt{-3}$ is generated by one inertia group, the member then stabilizes its ordinary line, a contradiction (\cref{sec:intro-uncond}).

For the coefficient $3$ the obstruction is the same curve $W$: the Catalan solution can be read as $2^3+1^3=3\cdot3$, a point of $x^3+y^3=3z^n$ with $n=1$, and $2^3+1^3=9\cdot1^n$ is a solution of $x^3+y^3=9z^n$ for every $n$.
\begin{maintheorem}\label{thm:main-3zn}
For every integer $n\ge3$, the equation $x^3+y^3=3z^n$ has no solution in coprime nonzero integers.
\end{maintheorem}
This is \cref{thm:3zp-family}, whose case $n=3$ is classical.  The prime exponents $p\equiv3\pmod4$ with $p\ge7$ come from Freitas's symplectic argument applied to a twist of Kraus's Frey curve, the prime exponents $7\le p<10^8$ from a criterion of Kraus's type checked by computer (\cref{thm:3zp-unconditional}), and the exponents $4$ and $5$ from descents in the manner of Bruin \cite{Bruin00} (\cref{thm:3zp-45}).  The prime exponents $p\ge17$ also come from the method of \cref{thm:main-uncond}, which for $p\equiv2\pmod 3$ is carried out at a prime that is inert in $K=\Q(\sqrt{-3})$ (\cref{thm:3zp-large}).  At such a prime the compatible system comes from a lifting lemma (\cref{lem:3zp-lift}), whose proof is that of a lifting theorem of Bellovin and Gee \cite[Theorem~5.2.1]{BellovinGee19} written out for this case.  \Cref{thm:main-uncond} does not use this lemma, and \cref{thm:main-3zn} depends on it only for the exponents $n\ge3$ that are divisible neither by $3$ nor by $4$ and whose prime factors $p\ge5$ all satisfy $p\equiv5\pmod{12}$ and $p>10^8$ (see the proof of \cref{thm:3zp-family}).  For $n=2$ there are solutions, such as $11^3+(-2)^3=3\cdot21^2$.  The equation $x^3+y^3=9z^4$, which besides $2^3+1^3=9\cdot1^4$ has the solution $71^3+(-23)^3=9\cdot14^4$, remains open (\cref{rem:3zp-open}).

\subsection{The Frey motive}
Let $K=\Q(\omega)=\Q(\sqrt{-3})$, where $\omega$ is a primitive cube root of unity, let $\OO=\Z[\omega]$ be its ring of integers and $\p=(\sqrt{-3})$ its prime above $3$, and let $\zeta_9$ be a primitive ninth root of unity; then $K(\zeta_9)=\Q(\zeta_9)$.  For $s\ne 0,1$ consider the affine surface
\[
Z_s\colon\ (1-y_0^9)(1-y_1^9)(1-y_2^9)=s
\]
with the action of $\mu_9^3$ by multiplication on the coordinates.  It has a twisted form $X_s$ over $K$, which becomes $Z_s$ over $\Kbar$ but on which $G_K$ also permutes the three coordinates cyclically, through $\Gal(\Q(\zeta_9)/K)$ (\cref{sec:motive}); without this permutation no character of order $9$ on each factor of $\mu_9^3$ would be fixed by $G_K$.  A character $\chi$ of $\mu_9^3$ that has order $9$ on each factor and is fixed by $G_K$ cuts out of the cohomology $H^2_c$ of the surfaces $X_s$ a local system $\VV$ of rank three on $\PP^1\smallsetminus\{0,1,\infty\}$ over~$K$.  Geometrically $\VV$ is the hypergeometric local system with local exponents $\{1/9,4/9,7/9\}$ (up to a common sign) at $s=\infty$ and $\{0,0,0\}$ at $s=0$.  Its local monodromy is unipotent with one Jordan block at $0$ (so $s=0$ is a point of maximally unipotent monodromy), is a pseudo-reflection of order $3$ at~$1$, and is semisimple with eigenvalues $\zeta_9,\zeta_9^{4},\zeta_9^{7}$, or their inverses, at~$\infty$, so that its cube is a scalar.  The field $K$ is forced by the exponents: $\{1/9,4/9,7/9\}$ is stable under multiplication by $4$ and $7$ modulo $1$, but not by $-1$, so the local system is defined over the fixed field $\Q(\sqrt{-3})$ of $\{1,4,7\}\subset(\Z/9)^\times$ and not over $\Q$.

To a solution $(a,b,c)$ of $x^3+y^3=z^p$ we attach the parameter
\[
s_0=\frac{c^p}{u^3}=1+\Big(\frac{w}{u}\Big)^3,\qquad \{u,w\}=\{a,b\},\ p\nmid u,
\]
and the \emph{Frey representation} $\rho=\VV_{s_0}\otimes\kappa_u^{-\ninf}$ of $G_K$ (after Darmon's Frey representations \cite{Darmon00}), where $\kappa_u$ is the cubic Kummer character of $u$.  Here $\ninf\in\{1,2\}$ is given by the local monodromy at $\infty$, which factors through a quotient $\mu_9$ of the inertia group there: its cube is a scalar, so the subgroup $\mu_3\subset\mu_9$ acts on $\VV$ by scalars, $\zeta\mapsto\zeta^{\ninf}$ (\cref{prop:geometry}).  The reduction of $\rho$ modulo $p$ is the \emph{residual Frey representation} $\rhob$.  Let $\q\nmid 3p$ be a prime of $K$ dividing $abc$.  If $\q\mid c$, then $v_\q(s_0)=p\,v_\q(c)\equiv0\pmod p$, and inertia at $\q$ acts through a power of the unipotent monodromy at $0$ with exponent divisible by $p$, which is trivial modulo $p$.  If $\q\mid w$, then $v_\q(s_0-1)=3v_\q(w)\equiv0\pmod 3$, and inertia acts through a power of the pseudo-reflection of order $3$ at $1$ with exponent divisible by $3$, which is trivial.  If $\q\mid u$, then $v_\q(s_0)=-3v_\q(u)\equiv0\pmod 3$, and inertia acts through a power of the monodromy at $\infty$ with exponent divisible by $3$, a scalar that the twist removes.  So $\rhob$ is unramified outside $3p$; it is conjugate self-dual up to the twist by the inverse square of the mod~$p$ cyclotomic character, and at the primes above $p$ it is the reduction of a crystalline representation with Hodge--Tate weights $\{0,1,2\}$ (\cref{sec:frey}).  It is irreducible (\cref{sec:irreducibility}).

\subsection{Why the obstruction disappears}
For Kraus's curve the separation fails at~$3$: both $E_{a,b}$ and $W$ have potentially multiplicative reduction at $3$, with Tate parameters of valuations $2p\,v_3(c)-3$ and $1$, both prime to $p$, so their mod~$p$ representations have the same shape at~$3$.  The rank-three motive is more sensitive.  Its parameter at a solution is $3$-adically very close to the maximally unipotent point: $v_3(s_0)=p\,v_3(c)\ge p$, because $3\mid c$ by a theorem of Kraus.  We prove (\cref{thm:inertia3}) that inertia at the prime $\p$ acts unipotently on $\VV_s$ for every $s$ in the punctured disc $v_3(s)>15/2$.  The radius has a simple origin: $\VV$ is a triple multiplicative convolution of Kummer sheaves of $1-x$ of order $9$, the binomial series of $(1-x)^{1/9}$ converges exactly on $v_3(x)>5/2$, and $15/2=3\cdot 5/2$.  The Catalan solution $1^3+2^3=3^2$ sits at $s=9$, with $v_3(s)=2$: outside the disc.

Two further ingredients of Part~I are computations at the two primes of $K$ above~$7$.  The irreducibility of $\rhob$ comes from the Frobenius polynomials at these primes of the fiber $\VV_2$ of $\VV$ at $s=2$: Kraus's congruence forces $7\nmid abc$ and $s_0\equiv 2\pmod 7$ (\cref{prop:kraus}~(c), \cref{lem:s0}), and $7$ is the smallest prime $\ell$ at which it confines $s_0$ to a single point of $\PP^1(\F_\ell)$ other than $0$, $1$ and $\infty$.  And the unipotence at $\p$ needs, after the geometry, one arithmetic input: the character by which $G_K$ acts on the line of invariants of the local monodromy in the fiber of $\VV$ at the tangential base point $\partial/\partial s$ at $s=0$ must be trivial.  This character has order dividing $3$ and is unramified outside $\p$, and a cubic character of $K_\p^\times$ is trivial on $1+\p^4$, where the binomial series of $(1+x)^{1/3}$ converges.  So it is a character of the ray class group of $K$ modulo $\p^4$, and again its triviality is decided at the two primes above~$7$ (\cref{sec:limit}).

\Cref{thm:inertia3} is a statement about $\VV$ alone, not about the equation, and it also says how the decomposition group at $\p$ acts: up to semisimplification, through $1$, $\varepsilon_\ell^{-1}$ and $\varepsilon_\ell^{-2}$, for every prime $\ell\ne3$.  At a prime $\q\nmid3\ell$, inertia acts unipotently on $\VV_s$ for every $s\in K$ with $v_\q(s)>0$, by tame ramification (\cref{lem:tame}); but $\p$ divides the denominator $9$ of the exponents of $\VV$.  The description of inertia for hypergeometric motives of rank two by Golfieri Madriaga, Pacetti and Rodriguez-Villegas excludes such primes, and they write that ``it would be interesting to have an explicit description for wild primes'' \cite[\S1]{GPRV26}.  The theorem is proved in \cref{sec:inertia3} without a semistable model.  Part~IV uses it again, for the solutions of $x^3+y^3=3^kz^p$ with $3\mid z$ or $k\ge8$ (\cref{prop:3zp-frey}).  We do not know whether the radius $15/2$ is optimal; for the Legendre family at $2$ and the Hesse family at $3$, convolutions of two Kummer sheaves, the same argument gives exactly the part near~$0$ of the locus of potentially multiplicative reduction (\cref{rem:radius}).

\subsection{The proof of Theorem~A}\label{sec:intro-uncond}
Let $p\ge17$ be a prime with $p\equiv1\pmod3$, and suppose that a solution exists.  By a theorem of Calegari, Emerton and Gee \cite[Theorem~4.2.11]{CEG22}, $\rhob$ lifts to a representation that belongs to a \emph{compatible system} $\{R_\lambda\}$: a family of three-dimensional representations $R_\lambda$ of $G_K$, the \emph{members} of the system, one for each prime $\lambda$ of a number field, with the same characteristic polynomials of Frobenius at almost all primes.  The system can be chosen polarized, with Hodge--Tate weights $\{0,1,2\}$, and with minimal ramification: each member is unramified outside $\p$ and the primes above its residue characteristic (Part~II).  At $\p$ the members have, up to Frobenius semisimplification, a common Weil--Deligne representation, on which inertia acts through a character of order dividing $3$ and whose monodromy is a nilpotent matrix $N$ (\cref{prop:members,prop:three-wd}).  The member at a prime above $p$, the \emph{$p$-adic member}, lifts $\rhob$.  Step~(1) uses the members at the primes above $5$, and steps~(3)--(5) the member $R$ at a prime above $3$ (Part~III), which is unramified outside $\p$ because $\p$ is the only prime of $K$ above $3$.  The proof has five steps.
\begin{enumerate}[(1)]
\item \emph{$N\ne0$, and $\rhob(I_\p)$ is not scalar.}  If $N$ were zero, inertia at $\p$ would act through scalars on the reductions modulo $5$ of the members at the primes above $5$.  A reduction that is a sum of three characters is excluded by the class numbers of $K(\sqrt5)$ and $K(\zeta_5)$: the characters would be $1$, $\epsf^{-1}$, $\epsf^{-2}$, with $\epsf$ the cyclotomic character modulo~$5$, and a nonsplit extension of $1$ or $\epsf^{-1}$ by $\epsf^{-2}$, given by a suitable lattice, would define an unramified cyclic extension of degree~$5$ of $K(\sqrt5)$ or $K(\zeta_5)$, the field cut out by the quotient of the two characters.  Otherwise the projective image of a constituent of dimension at least two would cut out a Galois extension of $K$ unramified outside $5$ with root discriminant below $3^{1/2}\cdot5^{3/2}<19.37$: the factor $3^{1/2}$ comes from the tame ramification of $K$ at $3$, and the factor $5^{3/2}$ from the ramification bound of Fontaine and Abrashkin at $5$ for Hodge--Tate weights in $[0,2]$ (\cref{prop:ram}).  The prime $5$ is the smallest prime $\ell$ for which these weights lie in the Fontaine--Laffaille range $[0,\ell-2]$.  When $25$ or $5$ does not divide the degree of the extension over $K$, its wild inertia groups at $5$ have order at most $5$, respectively $1$, and the same bound improves to $3^{1/2}\cdot5^{7/5}$, respectively $3^{1/2}\cdot5$; this excludes the smaller projective images.  Odlyzko's unconditional lower bounds for discriminants, class field theory and finite group theory exclude such an extension (part~(2) of \cref{thm:cases}).

The theorem of Calegari, Emerton and Gee also lets one prescribe an irreducible component of the local lifting ring at the place $3$ (a \emph{polarized component at $3$}, \cref{sec:systems}) on which the $p$-adic member must lie, and the argument just given applies whichever component is prescribed.  If $\rhob(I_\p)$ consisted of scalars, $\rhob$ would have a lift at the place $3$ with finite image on inertia, built with Hensel's lemma and Edixhoven's theorem on fixed points; this lift lies on a polarized component at $3$ that consists of lifts with $N=0$, and the system built on it would contradict what was just shown (\cref{prop:notscalar}).  So $\rhob(I_\p)$ is not scalar.
\item \emph{$N$ is a single Jordan block.}  As $\p$ is ramified over $3$, a generator of tame inertia at $\p$ can be taken to be the square of an element of $G_\Q$ outside $G_K$, and the polarization of $\rhob$ writes its image as a scalar times a matrix $A\,{}^tA^{-1}$ (\cref{prop:regular}).  A matrix of this form that is a scalar times a unipotent matrix is $1$ or a single Jordan block, because $A-{}^tA$ is alternating and so has even rank (\cref{lem:cosquare}).  Since $\rhob(I_\p)$ is not scalar (step~(1)), the unipotent factor of $\rhob$ on $I_\p$ is not $1$, so it is a single Jordan block.  It is the reduction of the unipotent factor $\exp(t_pN)$ of the $p$-adic member, which lifts $\rhob$ ($t_p$ is the $p$-adic tame character), and $(\exp(t_pN)-1)^2$, whose reduction is not $0$, would vanish if $N^2$ did; so $N$ is a single Jordan block too (\cref{cor:regular}).
\item \emph{$R$ is irreducible and ordinary at $\p$.}  A semisimple representation of dimension three whose monodromy at $\p$ is a single Jordan block is irreducible (\cref{prop:three-irreducible}), and weak admissibility forces the decomposition group at $\p$ to stabilize a line (\cref{prop:three-ordinary}).
\item \emph{Every reduction of $R$ has trivial semisimplification} (\cref{cor:three-trivial}).  We bound the root discriminant of the field cut out by a reduction and compare it with Odlyzko's unconditional lower bounds.  On the decomposition group at $\p$ a reduction is, up to an unramified character, upper unitriangular.  Its entries next to the diagonal are two homomorphisms with values in the residue field, combinations of the cubic characters of $\Q_3(\sqrt{-3})$, and the cup product of the two vanishes.  These characters form a space of dimension $4$ over $\F_3$; they are trivial on $1+\p^4$, and their upper ramification breaks are at most $3$ (\cref{lem:three-kummer}).  The extension cut out by such a representation has largest upper ramification break at most $11/3$, the largest break that a Heisenberg extension of degree $27$ of $K_\p$ can have (\cref{prop:three-type,prop:three-allk}; this also follows from a theorem of Abrashkin, \cref{rem:three-type-lit}), but the bound on the root discriminant that this alone gives lies above all of Odlyzko's bounds.

What brings the bound down is the size of the ramification groups between the breaks $3$ and $11/3$.  On the ramification groups of upper index above $3$ the entries next to the diagonal vanish, and the corner entry depends linearly on the exterior product of the two homomorphisms.  As their cup product vanishes, this exterior product lies in the kernel of the linear form that the cup product induces on the exterior square of the space of cubic characters; the exterior square has dimension $6$, and the kernel dimension $5$.  So it suffices to compute, by Kummer theory, the Heisenberg extensions of degree $27$ for five suitably chosen planes of cubic characters on which the cup product vanishes and whose exterior products span this kernel (\cref{prop:three-type,prop:three-allk}).  This brings the bound below Odlyzko's bounds for fields of large degree, with more room when the inertia groups at $\p$ are small (\cref{prop:three-rd}).  So the degree of the field is bounded, and finite group theory shows that a reduction has no irreducible constituent of dimension $2$ or $3$.  Its constituents are therefore characters unramified outside $\p$, which are trivial (\cref{prop:three-characters}).
\item \emph{Local generation.}  The field $K$ has one prime above $3$ and class number one, so the Galois group of the maximal pro-$3$ extension of $K$ unramified outside $\p$ is generated by an inertia group at $\p$ (\cref{lem:local-generation}).  A representation unramified outside $\p$ whose reduction has trivial semisimplification factors through this pro-$3$ group.  So $R$ has the same image as its inertia group at $\p$, which stabilizes the ordinary line, and $R$ is~reducible: a~contradiction.
\end{enumerate}

\subsection{Earlier work}
The method of Part~II, a lift of minimal ramification in a compatible system followed by the study of a member at a small prime, is that of Khare and Wintenberger for two-dimensional representations of $G_\Q$ \cite{KhareWintenbergerAnnals09,KhareWintenberger09}; earlier instances of the switch to a small prime are due to Dieulefait \cite{Dieulefait04} and Wintenberger \cite{Wintenberger04}.  The situation at $5$ in step~(1) is that of the theorems of Fontaine and Abrashkin on schemes with good reduction everywhere (\cref{rem:precedents}).  The residual step~(4) has a two-dimensional analogue in \c{S}eng\"un's theorem that $G_K$ has no irreducible two-dimensional representation over $\Fbar_3$ unramified outside $3$ \cite[Theorem~C]{Sengun09}.  That the Galois group of the maximal pro-$3$ extension of $K$ unramified outside $\p$ is generated by an inertia group at $\p$ (step~(5)) is an elementary and well-known property of $\Q(\sqrt{-3})$ and the prime $3$, going back to Shafarevich and Koch \cite{Shafarevich63,Koch02} (see \cite[Remark~6.1.4]{Greenberg16}).  The same step appears in dimension two in the work of Freitas, Kraus and Siksek \cite[Theorem~9(b) and the proof of Theorem~1]{FreitasKrausSiksek20}, where the image of the Galois representation of a suitable quadratic twist of an elliptic curve with good reduction away from one prime is shown to equal the image of an inertia group (see also \cite{FreitasKrausSiksek21}).  Representations of $G_K$ unramified outside $\p$ with residual image a $3$-group also occur in work of Freitas and S\'anchez-Rodr\'iguez \cite{FreitasSanchez25}.

A new ingredient is the way the ramification of the reductions of the member $R$ at a prime above~$3$ is bounded: Fontaine--Laffaille theory is not available for $R$, and its place is taken by ordinarity at $\p$, which weak admissibility forces once the monodromy operator is regular, given the Hodge--Tate weights $\{0,1,2\}$ (steps~(3) and~(4)).  On its own, this argument shows that no semisimple representation $G_K\to\GL_3(\overline\Q_3)$ unramified outside $\p$ is de Rham at $\p$ with Hodge--Tate weights $\{0,1,2\}$ and regular monodromy there (\cref{prop:three-scope}).

The Frey representation is three-dimensional, of unitary type over an imaginary quadratic field, and comes from a motive of rank three that is not of $\GL_2$-type: the local monodromy of $\VV$ at $s=1$ is a pseudo-reflection of order~$3$ (\cref{prop:geometry}), which no twist of the symmetric square of a rank-two local system has.  Darmon suggested that Katz's hypergeometric motives might connect generalized Fermat equations to higher-dimensional Galois representations \cite[remark after Theorem~1.13]{Darmon00}.  Rank-two hypergeometric motives have been developed for generalized Fermat equations by Golfieri Madriaga and Pacetti \cite{GolfieriPacetti} and by Golfieri Madriaga, Pacetti and Rodriguez-Villegas \cite{GPRV26}, who list the extension of their results to hypergeometric motives of any rank among the questions they expect to address; see also the Frey abelian varieties of \cite{BCDF-I}.  Among the neighboring problems, the twisted equations $x^3+y^3=q^\alpha z^p$, $q\ge5$ prime, were treated by Bennett, Luca and Mulholland \cite{BennettLucaMulholland11} and by Bennett, Bruni and Freitas \cite{BennettBruniFreitas18}, and recently by Freitas and Mocanu for $q=5$ and certain primes $p$ \cite[Theorem~1.8]{FreitasMocanu25}.  The signatures $(r,r,p)$ with $r\in\{5,7,11\}$ were treated by Billerey, Chen, Dieulefait and Freitas \cite{BCDF-I,BCDF-II}, in part conditionally on Darmon's ``big image'' conjecture \cite[Conjecture~4.1]{Darmon00}, in the form of \cite[Conjecture~1.2]{BCDF-I}.  For certain totally real fields $F'$, among them every $\Q(\zeta_{3^m})^+$, Dieulefait and Golfieri Madriaga \cite[Theorems~1.4 and~1.5]{DieulefaitGolfieri25} prove that Fermat's equation $x^p+y^p+z^p=0$ has no nontrivial primitive solution in the ring of integers of $F'$ for every prime $p\equiv2\pmod3$ beyond a bound depending on $F'$, using, when $3\mid xyz$, a level-lowering argument at a prime above $3$.

The conclusion of \cref{thm:main-uncond} was also known to follow from hypotheses of a different kind.  Under Baker's explicit $abc$ conjecture in the form of Laishram and Shorey, the equation has no solution for any $n\ge3$ other than $n=4$ \cite[Theorem~4]{LaishramShorey12} (the exponent $4$, which that theorem leaves out, is settled unconditionally \cite[Theorem~1]{Bruin00}, \cite[Theorem~2]{ChenSiksek09}); the $abc$ conjecture implies that there are only finitely many solutions over all exponents together \cite[\S5.2]{DarmonGranville95}; and the Frey--Mazur conjecture (see \cite[Conjecture~2.1]{FreitasKraus22}) implies the conclusion for $p$ large, by Kraus's congruence.  Zhou claims to derive the conclusion from inter-universal Teichmüller theory in a preprint \cite{Zhou25}, whose arguments we have not checked.

\subsection{Published results used}\label{sec:intro-inputs}
Besides finite computations, the proof of \cref{thm:main-uncond} uses the following published results.  Part~IV cites those that the proof of \cref{thm:main-3zn} uses in addition.
\begin{itemize}
\item On the equation: Kraus \cite[Th\'eor\`eme~6.1, Proposition~6.3~(ii), Lemme~6.4]{Kraus98}, with the modularity of elliptic curves over $\Q$ \cite{BCDT01}; and, for the exponents they cover, Chen--Siksek \cite[Theorem~2]{ChenSiksek09} and Freitas \cite[Theorem~3]{Freitas16}.
\item On the motive (Part~I): Katz's theory of hypergeometric sheaves \cite{KatzESDE} and Beukers--Heckman \cite{BeukersHeckman89}; proper and smooth base change \cite{SGA4}; Deligne's theory of weights \cite{DeligneWeilI,DeligneWeilII} and integrality \cite{SGA7II}; Abhyankar's lemma and tame specialization \cite{SGA1}, with tangential base points \cite{Deligne89}; Berkovich's étale cohomology of analytic spaces \cite{Berkovich93}; the uniqueness of a polarizable variation of Hodge structure on an irreducible local system \cite{Deligne87} and the Hodge numbers of hypergeometric variations \cite{Fedorov18}; Fontaine--Laffaille theory \cite{FontaineLaffaille}, Tsuji's comparison theorems \cite{Tsuji99,Tsuji02} in Beilinson's form \cite{Beilinson13}, and Mokrane's weight spectral sequence \cite{Mokrane93} in Nakkajima's corrected form \cite{Nakkajima05}.
\item For the compatible system (Part~II): Calegari--Emerton--Gee \cite[Theorem~4.2.11; also Lemma~1.5.20, Definition~2.1.9 and Theorem~2.1.16]{CEG22}, which rests on the potential automorphy theorems of Barnet-Lamb--Gee--Geraghty--Taylor; these theorems \cite[Theorems~4.5.1 and~5.5.1]{BLGGT14}, which we also apply directly, and further results of the same paper \cite[Lemmas~1.4.2, 1.4.3 and~5.3.1]{BLGGT14}; Bellovin--Gee \cite[Theorem~3.3.7]{BellovinGee19}; Edixhoven's theorem on fixed points of tame group actions \cite[Proposition~3.4]{Edixhoven92}; local--global compatibility, including at the residue characteristic \cite[Theorem~2.1.1]{BLGGT14}, \cite[Theorem~1.1]{BLGGTlgc2} (which rests on \cite{BLGGTlgc1}; see also \cite[Theorem~1.1]{Caraiani14}); Fontaine's Weil--Deligne representation of a potentially semistable representation \cite{Fontaine94pst}; Patrikis--Taylor \cite[Theorem~1.7]{PatrikisTaylor15}; Clozel--Harris--Taylor \cite[Lemma~2.1.1]{CHT08}; and Serre's theorem that locally algebraic abelian representations come from algebraic Hecke characters \cite{SerreAbelian}.
\item At the prime $5$: Fontaine--Laffaille theory and the ramification bound of Fontaine and Abrashkin \cite{Abrashkin89,Fontaine93} (\cref{thm:FA}).
\item At the prime $3$ (Part~III): Berger's theorem that de Rham representations are potentially semistable \cite{Berger02}, Fontaine's theory of semistable representations \cite{Fontaine94,Fontaine94pst} and the theorem of Colmez and Fontaine \cite{ColmezFontaine00}.
\item For discriminants and finite groups: the explicit formula in Poitou's unconditional form \cite{Poitou77,Odlyzko90}; the list of the simple groups of small order and of their automorphism groups in the ATLAS \cite{ATLAS}, and Dickson's classification of the finite subgroups of $\PGL_2(\Fbar_3)$ \cite{Dickson01,Faber23}.
\end{itemize}
The computations, listed in \cref{app:computations} with the software used and the separately written programs that repeat them, are in exact arithmetic, apart from evaluations of integrals with certified error bounds that give lower bounds for discriminants.  The programs and their outputs are in \cite{Companion}.

\subsection{Organization}
\Cref{sec:conventions} fixes conventions.  Part~I (\cref{sec:motive,sec:inertia3,sec:frey,sec:irreducibility}) constructs $\VV$ and the Frey representation, with its local properties and its irreducibility.  Part~II (\cref{sec:lift,sec:bounds,sec:reductions}) constructs the compatible system and carries out step~(1).  Part~III (\cref{sec:three,sec:three-reductions}) carries out steps~(2)--(5) and proves \cref{thm:main-uncond}.  Part~IV (\cref{sec:3zp}) treats the equations $x^3+y^3=3^kz^n$ with $k\ge1$ and proves \cref{thm:main-3zn}.

\Cref{sec:remarks} discusses the obstructing point and the exponents $p\equiv2\pmod3$ and states three open questions, and \cref{app:computations} lists the computations.  The Supplement, a separate document distributed with this paper, whose items are numbered S1, S2, \dots, contains the computations for Part~IV and computations that check results proved in the text.