TAOCP 1.3.1 Exercise 2
Four adjacent bytes represent integers from $0$ through $16{,}777{,}215$ by the table in Section 1.
Section 1.3.1: Description of MIX
Exercise 2. [02] If a value to be represented within MIX may get as large as 99999999, how many adjacent bytes should be used to contain this quantity?
Verified: yes
Solve time: 35s
Four adjacent bytes represent integers from $0$ through $16{,}777{,}215$ by the table in Section 1.3.1. Since
$$ 99{,}999{,}999 > 16{,}777{,}215, $$
no representation using four adjacent bytes can accommodate all values up to $99{,}999{,}999$.
Five adjacent bytes represent integers from $0$ through $1{,}073{,}741{,}823$, and this interval contains $99{,}999{,}999$. Hence five adjacent bytes suffice, and fewer than five do not suffice.
$$ \boxed{5} $$