In this paper, among other results, we prove that the title equation has finitely many solutions when $x - y = lz$ and $l$ is a fixed integer $\ne 2$. Moreover, all solutions $(x, y, z)$ satisfy $l < z < l^2 / 2$, $1 < y < l^2 / 2$ and $l^2+1 < x < (l^3 + l^2) / 2$. As a consequence, we extend a result of Cao.

Let $\mathcal{M}$ be a maximal ideal of a commutative ring $R$ such that $\sharp(R / \mathcal{M}) < \infty$ and $\operatorname{char} R / \mathcal{M} \ne 2$. Denoting the $\mathcal{M}$-adic completion of $R$ by $R_{\mathcal{M}}$, we will show $H^1(g, SL_n(R_{\mathcal{M}}))$ vanishes for $g = \langle s \rangle$ acting on $SL_n(R_{\mathcal{M}})$ via $A^s = (A^{-1})^t$ where $t$ is the transpose operator.

Let $k$ be a real quadratic number field and $\mathfrak{o}_k$, $E$ the ring of integers and the group of units in $k$. Denote by $E_{\mathfrak{p}}$ a subgroup represented by $E$ of $(\mathfrak{o}_k / \mathfrak{p})^{\times}$ for a prime ideal $\mathfrak{p}$ in $k$. We report that for a given positive integer $a$, the set of prime ideals of degree 1 for which the residual index of $E_{\mathfrak{p}}$ is equal to $a$ has a density under the Generalized Riemann Hypothesis.

In the present paper we establish some new integral inequalities involving convex function as a certain extensions of Pachpatte's inequality by using a fairly elementary analysis.

In this paper, we study the discreteness criteria for subgroups of $U(1, n; \mathbf{C})$ in complex hyperbolic space $H^n_{\mathbf{C}}$. We prove that a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$ with condition A is discrete if and only if every two generator subgroup of $G$ is discrete. We also prove that if a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$ contains a sequence of distinct elements $\{g_m\}$ with $\operatorname{Card}(\operatorname{fix}(g_m) \cap \partial H^n_{\mathbf{C}}) \ne \infty$ and $g_m \rightarrow I$ as $m \rightarrow \infty$, then $G$ contains a non-discrete, nonelementary two generator subgroup.

We give examples of isotropic submanifolds in spheres, which are counter-examples to the result of [S].