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1 %$Header$
2
3 \chapter*{Glossary Of Terms}
4 \markboth{GLOSSARY OF TERMS}{GLOSSARY OF TERMS}
5
6 \label{cglo0}
7
8 \begin{vworktermglossaryenum}
9
10 \item \textbf{axiom}\index{axiom}
11
12 A statement used in the premises of arguments and assumed to be true
13 without proof. In some cases axioms are held to be self-evident, as in
14 Euclidian geometry, while in others they are assumptions put forward for
15 the sake of argument.
16 (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
17
18 \item \textbf{cardinality}\index{cardinality}
19
20 The cardinality of a set is the
21 number of elements in the set. In this work, the cardinality
22 of a set is denoted $n()$. For example,
23 $n(\{12,29,327\}) = 3$.
24
25 \item \textbf{coprime}\index{coprime}
26
27 Two integers that share no prime factors are \emph{coprime}.
28 \emph{Example:}
29 6 and 7 are coprime, whereas 6 and 8 are not.
30
31 \item \textbf{GMP}\index{GMP}
32
33 The \emph{G}NU \emph{M}ultiple \emph{P}recision library.
34 The GMP is an arbitrary-precision integer, rational number,
35 and floating-point library that places no restrictions on
36 size of integers or number of significant digits in floating-point
37 numbers. This
38 library is famous because it is the fastest of its
39 kind, and generally uses asymptotically superior algorithms.
40
41 \item \textbf{greatest common divisor (g.c.d.)}
42
43 The greatest common divisor of two integers is the largest
44 integer which divides both integers without a remainder.
45 \emph{Example:} the g.c.d. of 30 and 42 is 6.
46
47 \item \textbf{integer}\index{integer}\index{sets of integers}\index{Z@$\vworkintset$}%
48 \index{integer!Z@$\vworkintset$}\index{integer!sets of}
49
50 (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) An \emph{integer}
51 (pronounced \emph{IN-tuh-jer}) is a whole number
52 (not a fractional number) that can be positive, negative, or zero.
53
54 Examples of integers are: -5, 1, 5, 8, 97, and 3,043.
55
56 Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14,
57 0.09, and 5,643.1.
58
59 The set of integers, denoted $\vworkintset{}$, is formally defined as:
60
61 \begin{equation}
62 \vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}
63 \end{equation}
64
65 In mathematical equations, unknown or unspecified integers are
66 represented by lowercase, italicized letters from the
67 ``late middle'' of the alphabet. The most common
68 are $p$, $q$, $r$, and $s$.
69
70 \item \textbf{irreducible}
71
72 A rational number $p/q$ where $p$ and $q$ are coprime
73 is said to be \emph{irreducible}.
74 Equivalently, it may be stated that $p$ and $q$ share no prime factors
75 or that the greatest common divisor of
76 $p$ and $q$ is 1.
77
78 \item \textbf{KPH}
79
80 Kilometers per hour.
81
82 \item \textbf{limb}\index{limb}
83
84 An integer of a size which a machine can manipulate natively
85 that is arranged in an array to create a larger
86 integer which the machine cannot manipulate natively and must be
87 manipulated through arithmetic subroutines.
88
89 \item \textbf{limbsize}\index{limbsize}
90
91 The size, in bits, of a limb. The limbsize usually represents
92 the size of integer that a machine can manipulate directly
93 through machine instructions. For an inexpensive microcontroller,
94 8 or 16 is a typical limbsize. For a personal computer or
95 workstation, 32 or 64 is a typical limbsize.
96
97 \item \textbf{MPH}
98
99 Miles per hour.
100
101 \item \textbf{mediant}\index{mediant}
102
103 The mediant of two fractions $m/n$ and $m'/n'$ is the fraction
104 $\frac{m+m'}{n+n'}$ (see Definition
105 \cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}). Note that the
106 mediant of two fractions with non-negative integer components
107 is always between them, but not usually exactly at the
108 midpoint (see Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}).
109
110 \item \textbf{natural number}\index{natural number}\index{integer!natural number}%
111 \index{sets of integers}\index{N@$\vworkintsetpos$}%
112 \index{integer!N@$\vworkintsetpos$}\index{integer!sets of}
113
114 (Nearly verbatim from \cite{bibref:w:wwwwhatiscom})
115 A \emph{natural number}
116 is a number that occurs commonly and obviously in nature.
117 As such, it is a whole, non-negative number.
118 The set of natural numbers, denoted $\vworkintsetpos{}$,
119 can be defined in either of two ways:
120
121 \begin{equation}
122 \label{cglo0:eq0001}
123 \vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}
124 \end{equation}
125
126 \begin{equation}
127 \label{cglo0:eq0002}
128 \vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}
129 \end{equation}
130
131 In mathematical equations, unknown or unspecified natural numbers
132 are represented by lowercase, italicized letters from the
133 middle of the alphabet. The most common is $n$, followed by
134 $m$, $p$, and $q$.
135 In subscripts, the lowercase $i$ is sometimes used to represent
136 a non-specific natural number when denoting the elements in a
137 sequence or series. However, $i$ is more often used to represent
138 the positive square root of -1, the unit imaginary number.
139
140 \textbf{Important Note:} The definition above is reproduced nearly
141 verbatim from \cite{bibref:w:wwwwhatiscom}, and (\ref{cglo0:eq0001})
142 is supplied only for perspective. In this work, a natural
143 number is defined by (\ref{cglo0:eq0002}) rather than (\ref{cglo0:eq0001}).
144 In this work, the set of non-negative integers is denoted by
145 $\vworkintsetnonneg{}$ rather than $\vworkintsetpos{}$.\index{Z+@$\vworkintsetnonneg$}%
146 \index{integer!Z+@$\vworkintsetnonneg$}\index{integer!non-negative}
147
148 \item \textbf{postulate}\index{postulate!definition}
149
150 An axiom (see \emph{axiom} earlier in this glossary). The term is usually
151 used in certain contexts, e.g. Euclid's postulates or Peano's postulates.
152 (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
153
154 \item \textbf{prime number}\index{prime number!definition}
155
156 (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) A \emph{prime number}
157 is a whole number greater than 1, whose only two whole-number
158 factors are 1 and itself. The first few prime numbers are
159 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. As we proceed in the set of
160 natural numbers $\vworkintsetpos{} = \{ 1, 2, 3, \ldots{} \} $, the
161 primes become less and less frequent in general.
162 However, there is no largest prime number.
163 For every prime number $p$, there exists a prime number $p'$ such that
164 $p'$ is greater than $p$. This was demonstrated in ancient times by the
165 Greek mathematician \index{Euclid}Euclid.\index{prime number!no largest prime number}%
166 \index{Euclid!Second Theorem}
167
168 Suppose $n$ is a whole number, and we want to test it to see if it is prime.
169 First, we take the square root (or the 1/2 power) of $n$; then we round this
170 number up to the next highest whole number. Call the result $m$.
171 We must find all of the following quotients:
172
173 \begin{equation}
174 \begin{array}{rcl}
175 q_m & = & n / m \\
176 q_{m-1} & = & n / (m-1) \\
177 q_{m-2} & = & n / (m-2) \\
178 q_{m-3} & = & n / (m-3) \\
179 & \ldots{} & \\
180 q_3 & = & n / 3 \\
181 q_2 & = & n / 2 \\
182 \end{array}
183 \end{equation}
184
185 The number $n$ is prime if and only if none of the $q$'s, as
186 derived above, are whole numbers.
187
188 A computer can be used to test extremely large numbers to see if they are prime.
189 But, because there is no limit to how large a natural number can be,
190 there is always a point where testing in this manner becomes too great
191 a task even for the most powerful supercomputers.
192 Various algorithms have been formulated in an attempt to generate
193 ever-larger prime numbers. These schemes all have limitations.
194
195 \end{vworktermglossaryenum}
196
197 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
198
199 \noindent\begin{figure}[!b]
200 \noindent\rule[-0.25in]{\textwidth}{1pt}
201 \begin{tiny}
202 \begin{verbatim}
203 $HeadURL$
204 $Revision$
205 $Date$
206 $Author$
207 \end{verbatim}
208 \end{tiny}
209 \noindent\rule[0.25in]{\textwidth}{1pt}
210 \end{figure}
211
212 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
213 %
214 %End of file C_GLO0.TEX

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