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

\subsection{The equation}\label{sec:intro-equation}

For integers $p,q,r\ge2$ the \emph{generalized Fermat equation} of signature $(p,q,r)$ is
\[
x^p+y^q=z^r,\qquad \gcd(x,y,z)=1,\qquad xyz\ne0 .
\]
When $1/p+1/q+1/r<1$, Darmon and Granville~\cite{DarmonGranville} proved that it has only finitely many solutions, and the Fermat--Catalan conjecture predicts that, taken over all such signatures together, there are only finitely many solutions in positive integers (counting $1+2^3=3^2$ once).  Ten are known, and in each of them one exponent is $2$ (see~\cite{BMS} and the survey~\cite{Survey}).  This paper concerns the signature $(2,5,7)$.  Ratcliffe and Grechuk~\cite[\S1.1]{Survey} note that, among the signatures $(p,q,r)$ with $p$, $q$, $r$ prime and $1/p+1/q+1/r<1$, it is the one with the smallest sum $p+q+r$ for which the equation was open.

Signatures whose exponents are three distinct primes form one of the main families of open cases.  As Putz notes~\cite[p.~2]{Putz}, the only such signatures resolved when he wrote his thesis were $(2,3,7)$~\cite{PSS07} and, assuming the generalized Riemann hypothesis, $(2,3,11)$~\cite{FNS20}.  For $(3,5,7)$, Dahmen and Siksek~\cite{DahmenSiksek} resolve the equation under divisibility conditions, and the recent preprints of Chocian~\cite{Chocian26} and of the author~\cite{Jaswal357} treat the whole equation.  Both build on a classification due to Dahmen, Siksek and Putz, which the second states as a hypothesis.  The modular method needs a Frey curve, and Frey elliptic curves are available for the signatures $(2,3,p)$ over $\Q$ and $(q,q,r)$ over number fields~\cite[p.~3]{Putz}.  For other signatures Darmon's programme~\cite{Darmon00} would use Frey abelian varieties instead, but it depends on modularity results, and on large-image results for the associated Galois representations, that are not yet available~\cite[p.~2]{Putz}.  Putz instead works with a Belyi map, as we recall in \cref{sec:intro-putz}.

We prove that $x^2+y^5=z^7$ has no solution in nonzero coprime integers, assuming Putz's reduction of the problem to a single number field (\cref{thm:putz} below).  Because two of the three exponents are odd, a solution of any one of
\[
x^2+y^5=z^7,\qquad x^2+y^7=z^5,\qquad x^5+y^7=z^2
\]
gives a solution of the others after changing signs and moving a term across the equation (for instance, $x^2+y^7=z^5$ is $x^2+(-z)^5=(-y)^7$).  Following Putz~\cite{Putz}, we write the equation as
\begin{equation}\label{eq:main}
X^5+Y^2=Z^7,\qquad \gcd(X,Y,Z)=1,\qquad XYZ\ne0,
\end{equation}
so that $(X,Y,Z)=(y,x,z)$ for a solution of $x^2+y^5=z^7$.  A prime dividing two of $X,Y,Z$ divides the third, so $X,Y,Z$ are pairwise coprime.  We call a triple of integers satisfying~\eqref{eq:main} a \emph{solution}.  Signs are allowed.

\subsection{Putz's reduction}\label{sec:intro-putz}

Put
\begin{equation}\label{eq:psi}
\psi(t)=25t^3+20t^2+14t+14,\qquad \Psi(t)=10t^4+4t^3+2t^2+2t-1 .
\end{equation}
The rational function $\varphi(t)=4t^5\psi(t)/(4t-1)$ of degree $8$ satisfies $\varphi(t)-1=\Psi(t)^2/(4t-1)$, because
\begin{equation}\label{eq:belyi}
4t^5\psi(t)-(4t-1)=\Psi(t)^2 .
\end{equation}
So, as a map $\PP^1\to\PP^1$, $\varphi$ is ramified only above $0$, $1$ and $\infty$, with ramification indices $(5,1,1,1)$ above $0$, $(2,2,2,2)$ above $1$ and $(7,1)$ above $\infty$: it is a Belyi map adapted to the exponents $5$, $2$, $7$.  For a solution put
\begin{equation}\label{eq:eta}
\eta=\frac{X^5}{Z^7}\in\Q\smallsetminus\{0,1\},\qquad 1-\eta=\frac{Y^2}{Z^7},
\end{equation}
and let $\Aeta$ be the fibre of $\varphi$ above $\eta$, that is, the $\Q$-algebra
\begin{equation}\label{eq:Aeta}
\Aeta=\Q[t]/(\feta),\quad \feta(t)=4t^5\psi(t)-\eta(4t-1)=100t^8+80t^7+56t^6+56t^5-4\eta t+\eta .
\end{equation}
It has dimension $8$ and is \'etale (\cref{lem:disc}), and it is unramified at every prime other than $2$, $5$ and $7$~\cite[Proposition~4.11]{Putz}.  Following the method of Darmon and Granville, Putz~\cite[Chapter~4]{Putz} determines the possible completions of $\Aeta$ at $2$, $5$ and $7$, and at $3$, where $\Aeta$ is unramified but its possible completions still restrict the fields that can occur~\cite[Theorem~4.6]{Putz}.  He then excludes the cases in which $\Aeta$ is not a field or has a subfield of degree $2$ or $4$ (using, among other tools, Chabauty's method on elliptic curves), and enumerates the octic fields with the required local behaviour by targeted Hunter searches: enumerations of the polynomials whose coefficients satisfy the bounds that follow from Hunter's theorem and the congruences modulo powers of $2$, $3$, $5$ and $7$ that the required completions impose.  The polynomials within Hunter's bounds include a defining polynomial of every primitive number field of a given degree and discriminant.  The largest of these searches took about $46\,000$ hours of processor time~\cite[\S3.7.1]{Putz}.  The outcome is his Theorem~4.33 (\cref{thm:putz} below): for every solution, $\Aeta$ is isomorphic to the field
\begin{equation}\label{eq:L8}
\begin{gathered}
\Lo=\Q(a),\qquad h(a)=0,\\
h(x)=x^8+4x^7-28x^6-168x^5-140x^4+560x^3+840x^2-480x-940 .
\end{gathered}
\end{equation}
From this Putz derives congruence conditions on solutions, for example $5\mid X$ and the absence of solutions with $2\mid X$ or $4\mid Z$~\cite[Theorem~4.34 and Corollary~4.35]{Putz}.  He discusses three strategies for excluding $\Lo$ and leaves each open~\cite[\S4.5]{Putz}.  In a lecture in April 2025, Dahmen described partial resolutions for the signatures $(2,5,7)$ and $(3,5,7)$ under stringent congruence conditions, from work in part joint with Siksek and Putz~\cite{DahmenTalk}.

\subsection{Main results}

\Cref{thm:L8} below is our main unconditional result.  \Cref{thm:main}, which resolves the equation, combines it with Putz's theorem and is conditional on that theorem.

Let $b$ be a root of $\psi$ in an algebraic closure of $\Lo$.  The polynomial $\psi$ is irreducible over $\Lo$ (\cref{lem:fields}), so $\Lt=\Lo(b)$ is a field of degree $24$.

\begin{theorem}\label{thm:L8}
Let $(X,Y,Z)$ be a solution of~\eqref{eq:main}.  Then $\Aeta$ is not isomorphic to $\Lo$.
\end{theorem}

The proof uses the fact that the class number of $\Lt$ is $1$ (\cref{thm:classnumber}), which we prove in \cref{app:classnumber} without assuming the generalized Riemann hypothesis.  Together with Putz's theorem, \cref{thm:L8} gives:

\begin{theorem}\label{thm:main}
Assume that for every solution of~\eqref{eq:main} the algebra $\Aeta$ is isomorphic to $\Lo$, as Putz's Theorem~4.33~\cite{Putz} asserts \textup{(}\cref{thm:putz}\textup{)}.  Then the equation $x^2+y^5=z^7$ has no solution in nonzero coprime integers $x,y,z$.  Equivalently, none of the equations $x^2+y^5=z^7$, $x^2+y^7=z^5$ and $x^5+y^7=z^2$ has a solution in nonzero coprime integers.
\end{theorem}

\begin{proof}
By hypothesis, a solution of~\eqref{eq:main} has $\Aeta\isom\Lo$, which \cref{thm:L8} excludes.  The equivalent forms follow from \cref{sec:intro-equation}.
\end{proof}

\subsection{The idea of the proof}\label{sec:idea}

Suppose that $\Aeta\isom\Lo$, and let $\vt\in\Lo$ be the image of the class of $t$, so that $\feta(\vt)=0$, that is,
\begin{equation}\label{eq:theta}
4\vt^5\psi(\vt)=\eta\,(4\vt-1).
\end{equation}
Since $\eta=X^5/Z^7$, \eqref{eq:theta} says that $\vt^5\psi(\vt)=X^5(4\vt-1)/(4Z^7)$.  Over $\Lt$ the factor $\vt-b$ of $\psi(\vt)$ becomes available, and the descent studies it modulo fifth powers, through the element
\begin{equation}\label{eq:E}
E=80000\,(\vt-b)\,\psi(\vt)^3\in\Lt .
\end{equation}
The constant $80000=2^7\cdot5^4$ only normalizes $E$.  Its factor $5^4$ cancels, up to fifth powers, the denominator $25$ of $\Norm{\Lt/\Lo}{\vt-b}=\psi(\vt)/25$ (see~\eqref{eq:normpsi}), since $80000^3/25=2^{21}\cdot5^{10}$ (\cref{prop:D2}).  Its factor $2^7$ is a choice of normalization, with which the norm of $E$ is $2$ times a fifth power.  Any other power of $2$, or any other nonzero rational constant, would serve equally well: it would change the class of $E$ by a fixed class, and the conditions below accordingly.

The proof has four steps.
\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item \emph{Descent} (\cref{sec:descent}).  Because $X$ and $Z$ are coprime, the valuation of $E$ at every prime of $\Lt$ not above $2$, $5$ or $7$ is divisible by $5$ (\cref{prop:D1}).  So the class of $E$ lies in the finite group $\Sel$ of classes in $\Lt^\times/\Lt^{\times5}$ with this property, where $S$ is the set of primes of $\Lt$ above $2$, $5$ and $7$.  Moreover, for every $\vt\in\Lo$ the norm of $E$ to $\Lo$ is $2$ times a fifth power (\cref{prop:D2}).  This is the reason for the factor $\psi(\vt)^3$: up to a constant it is the cube of the norm of $\vt-b$, and for any element $u$ of a cubic extension, $u\cdot\Nm(u)^3$ has norm $\Nm(u)^{10}$, a fifth power.
\item \emph{Local conditions at $2$ and $5$} (\cref{sec:local}).  The field $\Lt$ has a single prime above $2$, and it is totally ramified (\cref{lem:labels}).  There the valuation of $E$ is $3$ modulo $5$ for every $\vt\in\Lo$, a consequence of the norm identity (\cref{lem:two}).  At the seven primes above $5$ the valuations of $E$ modulo $5$ are $2,3,0,0,0,0,0$ for every solution (\cref{prop:five}).  Because $\vt$ generates $\Lo$, the factorization of $\feta$ over $\Qp5$ must match the decomposition of $5$ in $\Lo$, and Newton polygons then determine the valuations.  The proof also shows $5\nmid YZ$.
\item \emph{Auxiliary primes} (\cref{sec:aux}).  If a prime $q\notin\{2,5,7\}$ divides $XYZ$, then $\eta$ reduces modulo $q$ to one of the branch points $0$, $\infty$, $1$ of $\varphi$, and the splitting of $\Aeta$ at $q$ is then of one of three restricted kinds, read off from the factorizations modulo $q$ of $\psi$, of binomials $s^5-c$ and $s^7-c$, and of $\Psi$ (\cref{prop:A1}).  A search of the primes $q\equiv1\pmod5$ up to $700$, for which $\Fq q$ contains the fifth roots of unity, finds sixteen at which the splitting of $\Lo$ is of none of these kinds, so that $q\nmid XYZ$.  Then the residues of $\vt$ at the primes of $\Lo$ above $q$ are roots of the reduction of $\feta$ modulo $q$, which depends only on $\eta\bmod q$.  Running through all values of $\eta\bmod q$ and all such roots gives a finite set $\Iq$ that must contain the vector of fifth-power residue symbols of $E$ at the primes of $\Lt$ above $q$ (\cref{prop:A2}).
\item \emph{The sieve} (\cref{sec:selmer,sec:sieve}).  Since the class number of $\Lt$ is $1$ (\cref{thm:classnumber}), $\Sel$ is the group of $S$-units of $\Lt$ modulo fifth powers.  It has dimension $24=r_1+r_2-1+\#S$ over $\F_5$, where $(r_1,r_2)=(2,11)$ is the signature of $\Lt$ and $\#S=12$, and \cref{tab:B} gives a basis of it consisting of $S$-units.  The norm condition cuts the possible classes of $E$ to an affine subspace of dimension $14$, and the valuations at the primes above $5$ cut it to dimension $10$.  The sieve then uses the conditions at the five auxiliary primes with the smallest proportion $\#\Iq/5^m$, where $m$ is the number of primes of $\Lt$ above $q$.  Taken in turn, $q=181,311,131,251,101$ leave $36\,875$, $567$, $19$, $1$ and $0$ classes (\cref{thm:sieve}).
\end{enumerate}
The last step is a finite computation over $\F_5$, and all of its data are printed in \cref{app:data}.

That five primes are enough is not surprising.  At an auxiliary prime $q$ the norm condition ties together the symbols of $E$ above each prime of $\Lo$, so on the affine subspace of dimension $10$ the vector of symbols at $q$ takes at most $5^{m-m'}$ values, where $m'$ is the number of primes of $\Lo$ above $q$.  At the five primes used it takes exactly $5^{m-m'}$ values, and the fraction of these that lie in $\Iq$ ranges from $59/5^6$ at $q=181$ to $16/5^4$ at $q=101$.  If the five conditions were independent, the expected number of surviving classes would be about $0.001$ (\cref{rem:whyfive}).

\Cref{sec:control} applies the same computation to the fibres of $\varphi$ above two rational points of the twisted equation $X^5+2Y^2=Z^7$ ($\eta=-1$ and $\eta=243$).  Each of these fibres is an octic field $K_8$ not isomorphic to $\Lo$, and over $K_{24}=K_8(b)$ the element defined as in~\eqref{eq:E} exists, so its class must satisfy every condition.  The sieve keeps exactly one class, the class of $E$.  This tests the conditions and their implementation on fibres where the answer is known.

\subsection{Relation to earlier work}\label{sec:related}

The field $\Lt$, and the idea of a factorization into fifth powers over it, are due to Putz.  In~\cite[\S4.5.2, pp.~150--152]{Putz} he considers the compositum of $\Lo$ with the cubic field $\Q[U]/(U^3+8U^2+56U+560)$.  Since $U^3+8U^2+56U+560=40\,\psi(U/10)$, this is $\Lt$, with $10b$ as the root of the cubic.  Putz writes $r$ for our $\vt$, and $\theta$ for $10b$.  For a solution with $\Aeta\isom\Lo$ he puts $w=10\vt Z$~\cite[Lemma~4.38]{Putz} and divides $w$ and $Z$ by their greatest common divisor in $\Lo$ to obtain $w_0,z_0$.  He shows that the cubic form $g(U,V)=U^3+8U^2V+56UV^2+560V^3$ satisfies $g(w_0,z_0)=\alpha y_0^5$, with $\alpha$ and $y_0$ in the ring of integers of $\Lo$ and $\alpha$ divisible only by primes above $2$, $5$ and $7$~\cite[Proposition~4.39]{Putz}.  This uses $5\mid X$ and the triviality of the class group of $\Lo$.  Assuming that the class group of $\Lt$ is trivial, and remarking that a class number prime to $5$ may suffice, he writes the linear factor $g_1(w_0,z_0)=w_0-10bz_0$ as $\alpha_1\beta_1^5$, with $\alpha_1$ divisible only by primes above $2$, $5$ and $7$.  He thereby reduces the problem to rational points on finitely many curves of genus $2$ over $\Lt$, a route he did not pursue.  Our element is closely related to his: $E=g_1(w,Z)\,g(w,Z)^3/(8Z^{10})$, so that, modulo fifth powers, $E$ is $4\,g_1(w_0,z_0)\,g(w_0,z_0)^3$ (\cref{rem:putzE}).

What is new here is the following.
\begin{enumerate}[label=(\roman*),leftmargin=2em]
\item The factor $\psi(\vt)^3$, which makes the norm of $E$ a known class, the class of $2$ in $\Lo^\times/\Lo^{\times5}$ (\cref{prop:D2}).  In Putz's normalization the norm of $g_1(w_0,z_0)$ is $g(w_0,z_0)=\alpha y_0^5$, whose class, that of $\alpha$, is unknown.
\item A direct proof that the class of $E$ lies in $\Sel$ (\cref{prop:D1}), which uses only that $X$ and $Z$ are coprime: it needs neither the class number of $\Lo$ nor $5\mid X$.
\item The valuations of $E$ at the primes above $2$ and $5$ (\cref{sec:local}), derived from the isomorphism $\Aeta\isom\Lo$ alone, without Putz's congruence conditions.
\item The use of auxiliary primes in the descent.  Putz used his local description of $\Aeta$ at primes dividing $XYZ$ (\cref{prop:A1}) to exclude $q\mid XYZ$ and restrict $\eta$ modulo $q$ for $q=11$ and $197$~\cite[Theorem~4.34(v),(vi)]{Putz}.  Here the same description restricts the fifth-power residue symbols of $E$ at sixteen primes, and a computation eliminates every class (\cref{sec:aux,sec:selmer,sec:sieve}).
\item A proof, without the generalized Riemann hypothesis, that $\Lt$ has class number $1$ (\cref{app:classnumber}), which Putz assumed in the approach recalled above.
\end{enumerate}
Our proof of \cref{thm:L8} is a descent of a familiar kind: the class of $E$ lies in a finite group, local conditions cut out a subset of it, and one shows that this subset is empty.  For the descent map on cyclic covers of the projective line see Poonen and Schaefer~\cite{PS97}.  In the partial descent of Siksek and Stoll~\cite{SS12} for $x^3+y^4+z^5=0$ and the two-cover descent of Bruin and Stoll~\cite{BS09}, sets of this kind are shown to be empty in order to prove that a curve has no rational points.  They also appear in the working paper of Dahmen and Siksek on $x^5+y^3=z^7$~\cite{DahmenSiksek}.  As in those descents, the norm class of $E$ is known in advance, so that the norm gives a fixed affine condition.  The same finishing method, a fifth-power descent completed by residue symbols at auxiliary primes, is used in our treatment of $x^5+y^3=z^7$~\cite[\S3]{Jaswal357}, for a septic field left by the reduction there.

\subsection{The inputs of the proof}\label{sec:assumed}

The proof of \cref{thm:main} rests on the following.
\begin{enumerate}[label=(\roman*),leftmargin=2em]
\item Putz's Theorem~4.33~\cite{Putz} (\cref{thm:putz}).  It is part of a doctoral thesis and has not appeared in a refereed journal.  Its proof rests on local computations and elliptic-curve Chabauty computations in \textsc{Magma}, and on Hunter searches run with a C++ program, none of which we have repeated.  Putz's \textsc{Magma} scripts for the proofs of his Chapter~4, with their outputs, are public~\cite{PutzEtale}, and so is his program for the Hunter searches~\cite{PutzHunter}.  The scripts do not ask \textsc{Magma} for class group bounds that assume the generalized Riemann hypothesis.  They compute $2$-Selmer groups, rank bounds and Mordell--Weil subgroups of elliptic curves over number fields, which rest on class group and unit computations, and neither the thesis nor the scripts say whether these were proved.  The repository of these scripts names \textsc{Magma} V2.28-8 as compatible, and Elsenhans and Voight report that for some time before V2.29 a bug disabled the saturation step with which \textsc{Magma}'s default class group computation confirms its result, although they consider it unlikely that a wrong answer was returned~\cite{EV25}.
\item A criterion of Belabas, Diaz y Diaz and Friedman~\cite[Theorem~5.1]{BDF08}, deduced from Weil's explicit formula~\cite{Weil52} in the form given by Poitou~\cite{Poitou}, under which the prime ideals of small norm generate the class group of a number field.  It assumes no hypothesis on the zeros of $L$-functions~\cite[\S5]{BDF08}.  With it and with computations of the kind described in (iii), we prove in \cref{app:classnumber} that the class number of $\Lt$ is $1$ (\cref{thm:classnumber}).
\item Explicit computations with PARI/GP~\cite{PARI} and Python.  Each is stated in the text as a numbered lemma, proposition, theorem or displayed identity, or as one of the steps (a)--(c) of \cref{app:cn-computation}.  The data of the sieve are printed in \cref{app:data}, and the coefficients used for \cref{thm:classnumber} in \cref{tab:aj}.  The generators of the $665\,770$ prime ideals of \cref{app:classnumber} are too many to print, and the deposited programs~\cite{Companion257} find each of them again and check it as in \cref{app:cn-computation}(c), in about an hour.  \Cref{app:computations} lists the computations and says which were repeated by separately written programs.
\end{enumerate}
\Cref{thm:L8,thm:main} have been formalized in Lean~4 with Mathlib~\cite{Lean257}, following the proof given here, with \cref{thm:sieve} checked from the data printed in \cref{app:data}.  The formal statements take two hypotheses: Putz's theorem, for \cref{thm:main} only, and the absence of elements of order~$5$ in the class group of $\Lt$, a consequence of \cref{thm:classnumber}, whose proof in \cref{app:classnumber} is not formalized.  The formal proofs use only Lean's standard axioms.

\subsection{Plan of the paper}

\Cref{sec:putz} states Putz's theorem and describes the fields $\Lo$ and $\Lt$.  \Cref{sec:descent} shows that the class of $E$ lies in $\Sel$ and computes its norm to $\Lo$, \cref{sec:local} computes the valuations of $E$ at the primes above $2$ and $5$, and \cref{sec:aux} gives the conditions at the auxiliary primes.  \Cref{sec:selmer} describes the group $\Sel$ and states the class number of $\Lt$, and \cref{sec:sieve} carries out the sieve and proves \cref{thm:L8}.  \Cref{sec:control} shows, on two fibres of $\varphi$ above points of a twisted equation, that the conditions keep a class that exists.  \Cref{app:data} prints the data, \cref{app:classnumber} proves that the class number of $\Lt$ is $1$, and \cref{app:computations} describes the computations.  \Cref{tab:notation} lists the main notation.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}>{$}l<{$}>{\raggedright\arraybackslash}p{0.78\textwidth}@{}}
\toprule
\text{Symbol} & Meaning \\
\midrule
X,Y,Z & a solution of~\eqref{eq:main}; $(X,Y,Z)=(y,x,z)$ for $x^2+y^5=z^7$ \\
\eta & $X^5/Z^7$ \\
\psi,\ \Psi & $25t^3+20t^2+14t+14$ and $10t^4+4t^3+2t^2+2t-1$, with $4t^5\psi-(4t-1)=\Psi^2$ \\
\feta,\ \Aeta & $4t^5\psi(t)-\eta(4t-1)$; the algebra $\Q[t]/(\feta)$ \\
h,\ a,\ \Lo & the octic polynomial~\eqref{eq:L8}; a root of $h$; the field $\Q(a)$ \\
\vt & the root of $\feta$ in $\Lo$ given by an isomorphism $\Aeta\isom\Lo$ \\
b,\ \Lt & a root of $\psi$; the field $\Lo(b)$ of degree $24$ \\
E & $80000(\vt-b)\psi(\vt)^3\in\Lt$ \\
S & the twelve primes $\PT_1,\dots,\PT_{12}$ of $\Lt$ above $2$, $5$, $7$ (\cref{tab:Sprimes}) \\
\Sel & classes in $\Lt^\times/\Lt^{\times5}$ with valuation divisible by $5$ at every prime outside $S$ \\
B_1,\dots,B_{24} & the basis of $\Sel$ (\cref{prop:basis}, \cref{tab:B}) \\
u_1,\dots,u_{10} & $S$-units of $\Lo$ used for the norm condition (\cref{tab:u}) \\
\NM,\ \TT & the norm matrix and the coordinate vector of $2$ (\cref{sec:normdata}) \\
\cQ & the sixteen auxiliary primes (\cref{prop:A3}) \\
\sym_{\qQ} & the fifth-power residue symbol at a prime $\qQ$ of $\Lt$, with values in $\F_5$ \\
\Iq,\ \Mq & the allowed symbol vectors of $E$ at $q$; the symbols of $B_1,\dots,B_{24}$ at $q$ \\
\bottomrule
\end{tabular}
\vspace{3pt}
\caption{Main notation.}\label{tab:notation}
\end{table}