Let $q$ be an odd prime. Let $c>1$ and $t$ be positive integers such that $q^{t}+1=2c^{2}$. Using elementary method and a result due to Ljunggren concerning the Diophantine equation $\frac{x^{n}-1}{x-1}= y^{2}$, we show that the Diophantine equation $x^{2}+q^{m}=c^{2n}$ has the only positive integer solution $(x, m, n)=(c^{2}-1, t, 2)$. As applications of this result some new results on the Diophantine equation $x^{2}+q^{m} = c^{n}$ and the Diophantine equation $x^{2}+(2c-1)^{m} = c^{n}$ are obtained. In particular, we prove that Terai’s conjecture is true for $c=12,24$. Combining this result with Terai’s results we conclude that Terai’s conjecture is true for $2 \leq c \leq 30$.

In this paper we prove a well posed and an ill posed result in the Gevrey category for a simple model hyperbolic operator with triple characteristics, when the principal symbol cannot be smoothly factorized, and whose propagation cone is not transversal to the triple characteristic manifold, thus confirming the conjecture that the Ivrii-Petkov condition is not sufficient for the $C^{\infty}$ well posedness unless the propagation cone is transversal to the characteristic manifold, albeit for a limited class of operators. Moreover we are able not only to disprove $C^{\infty}$ well posedness, but we can actually estimate the precise Gevrey threshold where well posedness will cease to hold.

In this paper, we shall prove, for any $m\geq 1$, the existence of an uncountable subset of $U$-numbers of type $\leq m$ (which we called the set of \textit{$m$-ultra numbers}) for which there exists uncountably many transcendental analytic functions mapping it into Liouville numbers.

Let $G$ be a real reductive Lie group and $H$ a closed subgroup of $G$ which is reductive in $G$. In our earlier work it was shown that, if the homomorphism $i : H^{\bullet}(\mathfrak{g}_{\mathbf{C}}, \mathfrak{h}_{\mathbf{C}}; \mathbf{C}) \to H^{\bullet}(\mathfrak{g}_{\mathbf{C}},(\mathfrak{k}_{H})_{\mathbf{C}}; \mathbf{C})$ is not injective, there does not exist a compact manifold locally modelled on $G/H$. In this paper, we give a classification of the semisimple symmetric spaces $G/H$ for which $i$ is not injective. We also study the case when $G$ cannot be realised as a linear group.