On the Gaps between Two Consecutive Prime Numbers
Hisanobu Shinya
math0506196.tex
\title{On the Gaps between Two Consecutive Prime Numbers}
\author{Hisanobu Shinya}
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\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{scolium}{Scolium}  
\newtheorem{definition}{Definition}



\begin{document}
\newpage
\maketitle

Let $p_{i}$ be the $i$th prime number and 
\begin{displaymath}
d_{i} = p_{i + 1} - p_{i}.
\end{displaymath}

In this article, we establish a certain relation between $d_{k}$ and divisors of composite numbers between
$p_{k}$ and $p_{k+1}$.

Let $d_{k}$ be given.
For each natural number $1 \leq i < d_{k}$, there exists a prime number $p_{j}$ such that
\begin{equation} \label{A}
p_{k} \equiv -i \mod p_{j}.
\end{equation}
Thus, it is plain that $d_{k}$ is the \emph{smallest} natural number for which the congruence
\begin{equation} \label{B}
p_{k} + d_{k} \equiv r \mod p_{h}
\end{equation}
holds for each $h \leq k$ and a nonzero integer $r$.
Now, from (\ref{A}), we have
\begin{equation} \label{C}
\frac{p_{k} + i}{p_{j}}\in \mathbb{Z},
\end{equation}
where $i$ and $j$ are as defined above and $\mathbb{Z}$ is the set of all integers, and from (\ref{B}), 
\begin{equation} \label{D}
\frac{p_{k} + d_{k}}{p_{h}} \not\in \mathbb{Z},
\end{equation}
where $h$ can be any natural number less than or equal to $k$ as mentioned above.
Hence, if we choose $h = j$ in (\ref{D}) and combine (\ref{C}) and (\ref{D}), we have for each integer $z$
and each $i$,
\begin{displaymath}
\frac{p_{k} + i}{p_{j}}
\not= 
\frac{p_{k} + d_{k}}{p_{j}} + z
\end{displaymath}
or
\begin{equation} \label{E}
d_{k} \not= i - p_{j}z.
\end{equation}

This is the relation. Finally, it may be important to note that $p_{j}$ is dependent of $i$.

\end{document}
