From 4f670b750af5c11e1eac16d9cd8556455f89f46a Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan@mnpi.eu>
Date: Fri, 16 May 2025 22:18:44 +0200
Subject: Changed layout for more manageable volumes

---
 vol2/3.2.1.3.lyx | 273 +++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 273 insertions(+)
 create mode 100644 vol2/3.2.1.3.lyx

(limited to 'vol2/3.2.1.3.lyx')

diff --git a/vol2/3.2.1.3.lyx b/vol2/3.2.1.3.lyx
new file mode 100644
index 0000000..e94ce2c
--- /dev/null
+++ b/vol2/3.2.1.3.lyx
@@ -0,0 +1,273 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input defs
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children no
+\language english
+\language_package default
+\inputencoding utf8
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\float_placement class
+\float_alignment class
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style english
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tablestyle default
+\tracking_changes false
+\output_changes false
+\change_bars false
+\postpone_fragile_content false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+exerc1[M10]
+\end_layout
+
+\end_inset
+
+Show that,
+ no matter what the byte size 
+\begin_inset Formula $B$
+\end_inset
+
+ of 
+\family typewriter
+MIX
+\family default
+ happens to be,
+ the code (3) yields a random number generator of maximum period.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+This generator has 
+\begin_inset Formula $a=B^{2}+1$
+\end_inset
+
+,
+ 
+\begin_inset Formula $c=1$
+\end_inset
+
+,
+ and 
+\begin_inset Formula $m=B^{5}$
+\end_inset
+
+.
+ Since the prime divisors of 
+\begin_inset Formula $B^{5}$
+\end_inset
+
+ and 
+\begin_inset Formula $B^{2}$
+\end_inset
+
+ are the same,
+ the conditions of Theorem 3.2.1.2A are satisfied.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+exerc2[10]
+\end_layout
+
+\end_inset
+
+What is the potency of the generator represented by the 
+\family typewriter
+MIX
+\family default
+ code (3)?
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+3.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+rexerc6[20]
+\end_layout
+
+\end_inset
+
+Which of the values of 
+\begin_inset Formula $m=w\pm1$
+\end_inset
+
+ in Table 3.2.1–1 can be used in a linear congruential sequence of maximum period whose potency is 4 or more?
+ (Use the result of exercise 5.)
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+By exercise 5,
+ for a modulus 
+\begin_inset Formula $m=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}$
+\end_inset
+
+ with 
+\begin_inset Formula $e_{1}\geq\dots\geq e_{t}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $a=p_{1}+1$
+\end_inset
+
+ has the maximum potency,
+ which is 
+\begin_inset Formula $e_{1}$
+\end_inset
+
+,
+ except that if 
+\begin_inset Formula $m=p_{1}$
+\end_inset
+
+,
+ then the maximum potency is 0,
+ corresponding to 
+\begin_inset Formula $a=1$
+\end_inset
+
+.
+ Thus the values from the table that can be used are 
+\begin_inset Formula $10^{9}-1=3^{4}\cdot37\cdot333667$
+\end_inset
+
+ and 
+\begin_inset Formula $2^{27}+1=3^{4}\cdot19\cdot87211$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
-- 
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