Concrete mathematics : a foundation for computer science

4 EXERCISES 147
Why is “Euler” 32 Prove Euler’s theorem (4.50) by generalizing the proof of (4.47).
pronounced “Oiler”
when “Euclid” is
“Yooklid”?
33 Show that if f(m) and g(m) are multiplicative functions, then so is
h(m) = tdim f(d) g(m/d).
34 Prove that (4.56) is a special case of (4.61).
Homework exercises
35 Let I(m,n) be a function that satisfies the relation
I(m,n)m+ I(n,m)n = gcd(m,n),
when m and n are nonnegative integers with m # n. Thus, I( m, n) = m’
and I(n, m) = n’ in (4.5); the value of I(m, n) is an inverse of m with
respect to n. Find a recurrence that defines I(m,n).
36 Consider the set Z(m) = {m + n&? 1 integer m,n}. The number
m + no is called a unit if m2 - 1 On2 = f 1, since it has an inverse
(that is, since (m+nm).+(m-n&?) = 1). For example, 3+mis
a unit, and so is 19 - 6m. Pairs of cancelling units can be inserted into
any factorization, so we ignore them. Nonunit numbers of Z(m ) are
called prime if they cannot be written as a product of two nonunits. Show
that2,3,and4fnareprimesofZ(fl). Hint: If2=(k+L&?)x
(m + n&? ) then 4 = (kz - 1 012) ( mz - 1 On’). Furthermore, the square
of any integer mod 10 is 0, 1, 4, 5, 6, or 9.
37 Prove (4.17). Hint: Show that e, - i = (e,_l - i)’ + $, and consider
2-nlog(e, - t).
38 Prove that if a I b and a > b then
gcd(am _ bm, an _ bn) = agcd(m>n) _ bdm>ni , O$m