We consider the Diophantine equation as mentioned in the title and solve it completely, i.e., show that there exist no integer solution satisfying this equation.

In this note, we exhibit some examples of elliptic curves whose Mordell-Weil ranks grow in lower layer of the cyclotomic $\mathbf{Z}_p$-extension over the rationals.

Let $G$ be a discrete subgroup of $PU(1,n; \mathbf{C})$. For a boundary point $y$ of the Siegel domain, we define the generalized isometric sphere $I_y(f)$ of an element $f$ of $PU(1,n; \mathbf{C})$. By using the generalized isometric spheres of elements of $G$, we construct a fundamental domain $P_y(G)$ for $G$, which is regarded as a generalization of the Ford domain. And we show that the Dirichlet polyhedron $D(w)$ for $G$ with center $w$ convereges to $P_y(G)$ as $w \rightarrow y$. Some results are also found in [5], but our method is elementary.