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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass book
\begin_preamble
\input defs
\end_preamble
\use_default_options true
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\language english
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\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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