ardy and littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. they believed that the number $mathcal{r}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{r}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...

It was proved in [1] that for $k=4,5,6,7$ the elements of Somos-$k$ sequence defined by recurrence $$S_k(n+k)S_k(n)=\sum_{1\leqslant i\leqslant k/2}\alpha_i x_0\dots x_{k-1}S_k(n+k-i)S_k(n+i)$$ and initial values $S_k(j)=x_j$ ($j=0,\dots,k-1$) are polynomials variables $x_0,\dots,x_{k-1}$. The unit powers $x_j$ factors \linebreak $\alpha_i x_{k-1}$ can be reduced. In this paper, we find smalles...

We introduce a family of operators called the $k$-quasi-$(n,m)$-power normal operators. Such includes normal, $n$-normal and $(n,m)$-power An operator $T \in {\mathcal B}({\mathcal H})$ is said to be if it satisfies $$T^{*k}\bigg(T^nT^{*m}-T^{*m}T^n\bigg)T^k=0,$$ where $k,n$ $m$ are natural numbers. Firstly, some basic structural properties this established with help special kind matrix represe...