%$Header$
\chapter*{Glossary Of Terms}
\markboth{GLOSSARY OF TERMS}{GLOSSARY OF TERMS}
\label{cglo0}
\begin{vworktermglossaryenum}
\item \textbf{axiom}\index{axiom}
A statement used in the premises of arguments and assumed to be true
without proof. In some cases axioms are held to be self-evident, as in
Euclidian geometry, while in others they are assumptions put forward for
the sake of argument.
(Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
\item \textbf{cardinality}\index{cardinality}
The cardinality of a set is the
number of elements in the set. In this work, the cardinality
of a set is denoted $n()$. For example,
$n(\{12,29,327\}) = 3$.
\item \textbf{coprime}\index{coprime}
Two integers that share no prime factors are \emph{coprime}.
\emph{Example:}
6 and 7 are coprime, whereas 6 and 8 are not.
\item \textbf{GMP}\index{GMP}
The \emph{G}NU \emph{M}ultiple \emph{P}recision library.
The GMP is an arbitrary-precision integer, rational number,
and floating-point library that places no restrictions on
size of integers or number of significant digits in floating-point
numbers. This
library is famous because it is the fastest of its
kind, and generally uses asymptotically superior algorithms.
\item \textbf{greatest common divisor (g.c.d.)}
The greatest common divisor of two integers is the largest
integer which divides both integers without a remainder.
\emph{Example:} the g.c.d. of 30 and 42 is 6.
\item \textbf{integer}\index{integer}\index{sets of integers}\index{Z@$\vworkintset$}%
\index{integer!Z@$\vworkintset$}\index{integer!sets of}
(Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) An \emph{integer}
(pronounced \emph{IN-tuh-jer}) is a whole number
(not a fractional number) that can be positive, negative, or zero.
Examples of integers are: -5, 1, 5, 8, 97, and 3,043.
Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14,
0.09, and 5,643.1.
The set of integers, denoted $\vworkintset{}$, is formally defined as:
\begin{equation}
\vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}
\end{equation}
In mathematical equations, unknown or unspecified integers are
represented by lowercase, italicized letters from the
``late middle'' of the alphabet. The most common
are $p$, $q$, $r$, and $s$.
\item \textbf{irreducible}
A rational number $p/q$ where $p$ and $q$ are coprime
is said to be \emph{irreducible}.
Equivalently, it may be stated that $p$ and $q$ share no prime factors
or that the greatest common divisor of
$p$ and $q$ is 1.
\item \textbf{KPH}
Kilometers per hour.
\item \textbf{limb}\index{limb}
An integer of a size which a machine can manipulate natively
that is arranged in an array to create a larger
integer which the machine cannot manipulate natively and must be
manipulated through arithmetic subroutines.
\item \textbf{limbsize}\index{limbsize}
The size, in bits, of a limb. The limbsize usually represents
the size of integer that a machine can manipulate directly
through machine instructions. For an inexpensive microcontroller,
8 or 16 is a typical limbsize. For a personal computer or
workstation, 32 or 64 is a typical limbsize.
\item \textbf{MPH}
Miles per hour.
\item \textbf{mediant}\index{mediant}
The mediant of two fractions $m/n$ and $m'/n'$ is the fraction
$\frac{m+m'}{n+n'}$ (see Definition
\cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}). Note that the
mediant of two fractions with non-negative integer components
is always between them, but not usually exactly at the
midpoint (see Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}).
\item \textbf{natural number}\index{natural number}\index{integer!natural number}%
\index{sets of integers}\index{N@$\vworkintsetpos$}%
\index{integer!N@$\vworkintsetpos$}\index{integer!sets of}
(Nearly verbatim from \cite{bibref:w:wwwwhatiscom})
A \emph{natural number}
is a number that occurs commonly and obviously in nature.
As such, it is a whole, non-negative number.
The set of natural numbers, denoted $\vworkintsetpos{}$,
can be defined in either of two ways:
\begin{equation}
\label{cglo0:eq0001}
\vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}
\end{equation}
\begin{equation}
\label{cglo0:eq0002}
\vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}
\end{equation}
In mathematical equations, unknown or unspecified natural numbers
are represented by lowercase, italicized letters from the
middle of the alphabet. The most common is $n$, followed by
$m$, $p$, and $q$.
In subscripts, the lowercase $i$ is sometimes used to represent
a non-specific natural number when denoting the elements in a
sequence or series. However, $i$ is more often used to represent
the positive square root of -1, the unit imaginary number.
\textbf{Important Note:} The definition above is reproduced nearly
verbatim from \cite{bibref:w:wwwwhatiscom}, and (\ref{cglo0:eq0001})
is supplied only for perspective. In this work, a natural
number is defined by (\ref{cglo0:eq0002}) rather than (\ref{cglo0:eq0001}).
In this work, the set of non-negative integers is denoted by
$\vworkintsetnonneg{}$ rather than $\vworkintsetpos{}$.\index{Z+@$\vworkintsetnonneg$}%
\index{integer!Z+@$\vworkintsetnonneg$}\index{integer!non-negative}
\item \textbf{postulate}\index{postulate!definition}
An axiom (see \emph{axiom} earlier in this glossary). The term is usually
used in certain contexts, e.g. Euclid's postulates or Peano's postulates.
(Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
\item \textbf{prime number}\index{prime number!definition}
(Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) A \emph{prime number}
is a whole number greater than 1, whose only two whole-number
factors are 1 and itself. The first few prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. As we proceed in the set of
natural numbers $\vworkintsetpos{} = \{ 1, 2, 3, \ldots{} \} $, the
primes become less and less frequent in general.
However, there is no largest prime number.
For every prime number $p$, there exists a prime number $p'$ such that
$p'$ is greater than $p$. This was demonstrated in ancient times by the
Greek mathematician \index{Euclid}Euclid.\index{prime number!no largest prime number}%
\index{Euclid!Second Theorem}
Suppose $n$ is a whole number, and we want to test it to see if it is prime.
First, we take the square root (or the 1/2 power) of $n$; then we round this
number up to the next highest whole number. Call the result $m$.
We must find all of the following quotients:
\begin{equation}
\begin{array}{rcl}
q_m & = & n / m \\
q_{m-1} & = & n / (m-1) \\
q_{m-2} & = & n / (m-2) \\
q_{m-3} & = & n / (m-3) \\
& \ldots{} & \\
q_3 & = & n / 3 \\
q_2 & = & n / 2 \\
\end{array}
\end{equation}
The number $n$ is prime if and only if none of the $q$'s, as
derived above, are whole numbers.
A computer can be used to test extremely large numbers to see if they are prime.
But, because there is no limit to how large a natural number can be,
there is always a point where testing in this manner becomes too great
a task even for the most powerful supercomputers.
Various algorithms have been formulated in an attempt to generate
ever-larger prime numbers. These schemes all have limitations.
\end{vworktermglossaryenum}
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