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-#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
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-\index Index
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-\end_index
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-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-exerc1[10]
-\end_layout
-
-\end_inset
-
-Example (3) shows a situation in which 
-\begin_inset Formula $X_{4}=X_{0}$
-\end_inset
-
-,
- so the sequence begins again from the beginning.
- Give an example of a linear congruential sequence with 
-\begin_inset Formula $m=10$
-\end_inset
-
- for which 
-\begin_inset Formula $X_{0}$
-\end_inset
-
- never appears again in the sequence.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-This happens if 
-\begin_inset Formula $a=c=0$
-\end_inset
-
- and 
-\begin_inset Formula $0<X_{0}<m$
-\end_inset
-
-;
- then the sequence starts with 
-\begin_inset Formula $X_{0}\neq0$
-\end_inset
-
- and then it's constant at 0.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc2[M20]
-\end_layout
-
-\end_inset
-
-Show that if 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are relatively prime,
- the number 
-\begin_inset Formula $X_{0}$
-\end_inset
-
- will always appear in the period.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-For 
-\begin_inset Formula $a\geq2$
-\end_inset
-
-,
- by Euler's theorem,
- 
-\begin_inset Formula $a^{\varphi(m)}\equiv1\mod m$
-\end_inset
-
-,
- so 
-\begin_inset Formula $X_{\varphi(m)}=X_{0}$
-\end_inset
-
- by Equation (6).
- For 
-\begin_inset Formula $a=1$
-\end_inset
-
-,
- 
-\begin_inset Formula $X_{m}=(X_{0}+mc)\bmod m=X_{0}$
-\end_inset
-
-,
- and for 
-\begin_inset Formula $a=0$
-\end_inset
-
-,
- 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are not relatively prime.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-exerc3[M10]
-\end_layout
-
-\end_inset
-
-If 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are not relatively prime,
- explain why the sequence will be somewhat handicapped and probably not very random;
- hence we will generally want the multiplier 
-\begin_inset Formula $a$
-\end_inset
-
- to be relatively prime to the modulus 
-\begin_inset Formula $m$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-Let 
-\begin_inset Formula $p$
-\end_inset
-
- be a common prime between 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $c$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-X_{n}=(a^{n}X_{0}+(a^{n-1}+a^{n-2}+\dots+1)c)\bmod m,
-\]
-
-\end_inset
-
-so 
-\begin_inset Formula 
-\[
-X_{n}\bmod p=c\bmod p
-\]
-
-\end_inset
-
- for any 
-\begin_inset Formula $n\geq1$
-\end_inset
-
-.
-\end_layout
-
-\end_body
-\end_document