# Number Theory

## 1207 Submissions

 viXra:1207.0088 [pdf] submitted on 2012-07-24 13:05:32

### Differential Representation of Exact Value of the $n$th Partial Sum $\displaystyle \sum_{i=1}^{n}\frac{1}{a+(i-1)d}$ of General Harmonic Series

Authors: S. Maiti

In order to find a differential representation of the $n$th partial sum $\displaystyle \sum_{i=1}^{n}\frac{1}{a+(i-1)d}$ of the general harmonic series $\displaystyle \sum_{i=1}^{\infty}\frac{1}{a+(i-1)d}$, a theoretical study has been performed analytically. Moreover, some special cases of it such as harmonic number have been discussed.
Category: Number Theory

 viXra:1207.0087 [pdf] submitted on 2012-07-24 13:15:05

### Important Relations Over Some Summations $\left(\displaystyle \sum_{\substack{i,j=1\\(i<j)}}^{n}ij,~\sum_{\substack{i,j,k=1\\(i<j<k)}}^{n}ijk,~\sum_{\substack{i,j,k,l=1 \\(i<j<k<l)}}^{n}ijkl,~\cdots \right)$

Authors: S. Maiti

The paper is focused to find important relations and identities over some summations for natural numbers such as $\displaystyle \sum_{\substack{i,j=1\\(i<j)}}^{n}ij,~\sum_{\substack{i,j,k=1\\(i<j<k)}}^{n}ijk,~\sum_{\substack{i,j,k,l=1 \\(i<j<k<l)}}^{n}ijkl,~\cdots$. These relations are believed to find applications in the various branches of number theory particularly in the proposed theorems of Maiti \cite{Maiti1,Maiti2,Maiti3,Maiti4,Maiti5} which help to represent the factorial $n!$ entirely new way and also help to exhibit the $n$th partial sum of the general harmonic series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{a+(n-1) b}$ and its particular cases.
Category: Number Theory

 viXra:1207.0084 [pdf] replaced on 2012-07-30 16:45:30

### Counting the 'uncountable' Pushing the Boundaries in Number/set Theory

Authors: Salvatore Gerard Micheal
Comments: 9 Pages. author's email: micheal@msu.edu

Seven related papers develop a novel approach toward elucidating irrational density from a non-standard perspective. The notion of countability is explored in a precise new way. This new definition of countability clearly relates irrational to rational density and sets the stage for a more accessible understanding of the reals. Also, two implications in set theory are discovered. Constructive evaluation, criticism, and collaboration is invited.
Category: Number Theory

 viXra:1207.0083 [pdf] submitted on 2012-07-23 15:12:20

### New Expression of the Factorial of $n$ ($n!$, $n\in N$)

Authors: S Maiti
New Expression of the factorial of $n$ ($n!$, $n\in N$) is given in this article. The general expression of it has been proved with help of the Principle of Mathematical Induction. It is found in the form \begin{equation} 1+\sum_{i=1}^{n}a_i +\sum_{\substack{i,j=1 \\(i<j)}}^{n}a_ia_j + \sum_{\substack{i,j,k=1 \\(i<j<k)}}^{n}a_ia_ja_k +\cdots +a_1a_2\cdots a_{n}, \label{factorial_expression} \end{equation} where $a_i=i-1$ for $i=1,~2,~\cdots ,~n$. More convenient expression of this form is provided in Appendix.