Figurate primes and Hilbert's 8th problem

In this paper, by using the theory of elliptic curves, we discuss several Diophantine equations related with the so-called figurate primes. Meanwhile, we raise several conjectures related with figurate primes and Hilbert's 8th problem, including Goldbach's conjecture, twin primes conjecture and Catalan's conjecture as well.Comment: 7 pages, 1 figur

Quasi-Whittaker modules for the Schr\"odinger algebra

In this paper, we construct a new class of modules for the Schr\"{o}dinger algebra \mS, called quasi-Whittaker module. Different from \cite{[ZC]}, the quasi-Whittaker module is not induced by the Borel subalgebra of the Schr\"{o}dinger algebra related with the triangular decomposition, but its Heisenberg subalgebra \mH. We prove that, for a simple \mS-module $V$, $V$ is a quasi-Whittaker module if and only if $V$ is a locally finite \mH-module; Furthermore, we classify the simple quasi-Whittaker modules by the elements with the action similar to the center elements in U(\mS) and their quasi-Whittaker vectors. Finally, we characterize arbitrary quasi-Whittaker modules.Comment: 17 page