Comparison of Sin+ and Gen+ (n=2-15) Cationic Structures

We designed and optimized a large number of the isomers of Si12+ at the level of density functional theory (DFT)-B3LYP/6-311++G(d) using the Gaussian 03 software package. An unambiguous structure of the Si12+ cluster is presented, whose IR spectrum agrees well with the experiment result. The most stable geometric structures of Gen+(n=2-15) clusters were determined by the all-electron PBE/DND method in DMol3 of the Material Studio Package, and compared with those of the corresponding Sin+ geometries. Most structures of Gen+ and Sin+ are similar except the ones of those for n=9, 12, 13 and 14, and the pentagonal bipyramid motif and the tri-capped trigonal prism(TTP) motif often appear in the Sin+ and Gen+(n=7-15) structures(except for Si14+).     

关于Diophantine方程y2=px(x2+2)

对于Diophantine方程y2=px(x2+2),这里p为奇素数,证明了:当p=2593时,它有唯一的正整数解(x,y)=(72,31116).     

关于Diophantine方程y2=px(x2+2)的一点注记

设p是奇素数,运用初等数论方法证明了:如果P=16k4+1,这里k为正奇数,则方程y2=px(x2+2)无正整数解(x,y).