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\begin_layout Standard
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\backslash
exerc1[M10]
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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.
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answer 
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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.
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\backslash
exerc2[10]
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What is the potency of the generator represented by the 
\family typewriter
MIX
\family default
 code (3)?
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answer 
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3.
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\backslash
rexerc6[20]
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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.)
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answer 
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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

.
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