This paper presents some results on the simple exceptional Jordan algebra over an algebraically closed field $\Phi$ of characteristic not 2. Namely an example of simple decomposition of $H\left( {{O}_{3}} \right)$ into the sumof two subalgebras of the type $H\left( {{Q}_{3}} \right)$ is produced, and it is shown that this decomposition is the only one possible in terms of simple subalgebras.

Let $G\left( X \right)$ denote the number of positive composite integers $n$ satisfying $\sum\nolimits_{j=1}^{n-1}{{{j}^{n-1}}}\equiv -1\left( \,\bmod \,n \right)$. Then $G\left( X \right)\ll {{X}^{1/2}}\log \,X$ for sufficiently large $X$.

Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${{\text{R}}^{n}}$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total squaremean curvature $\int\limits_{M}{{{H}^{2}}d\sigma }$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$.