Concrete mathematics : a foundation for computer science

Too easy.
A more interesting
(still unsolved)
problem: Restrict
both cc and f~ to
be < 1 , and ask
when the given
multiset determines
the unordered
pair ia-, Bl.
A ANSWERS TO EXERCISES 499
Continuing along such lines now leads to the following interpretation:
K, is the least number > n in the multiset S of all numbers of the form
1 + a’ + a’ a2 + a’ a2a3 + . . . + a’ a2a3 . . . a, ,
where m 3 0 and each ok is 2 or 3. Thus,
S = {1,3,4,7,9,10,13,15,19,21,22,27,28,31,31,...};
the number 31 is in S “twice” because it has two representations 1 + 2 + 4 +
8 + 16 = 1 + 3 + 9 + l8. (Incidentally, Michael F’redman [108] has shown that
lim,,,K,/n = 1, ie., that S has no enormous gaps.)
3 44 Let diqi = DF!,mumble(q-l), so that DIP’ = (qD:_), +dp))/(q - 1)
and a$’ = ]D$‘,/(q -1)l. Now DF!, 6 (q - 1)n H a;’ < n, and the
results follow. (This is the solution found by Euler [94], who determined the
a’s and d’s sequentially without realizing that a single sequence De’ would
suffice.)
3.45 Let 01> 1 sati,sfy a+ I/R = 2m. Then we find 2Y, = a’” + aP2”, and
it follows that Y, = [a’“/21
3.46 The hint follows from (3.g), since 2n(n+ 1) = [2(n+ :)‘I. Let n+B =
(fi’ + fi’-‘)rn and n’ + 8’ = (fi”’ + &!‘)m, where 0 < 8,8’ < 1.
Then 8’ = 20 mod 1 = 28 - d, where d is 0 or 1. We want to prove that
n’ = Lfi(n + i )] ; this equality holds if and only if
0 < e/(2-JZ)+Jz(i -d) < 2.
To solve the recurrence, note that Spec( 1 + 1 /fi ) and Spec( 1 + fi ) partition
the positive integers; hence any positive integer a can be written uniquely in
the form a = \(&’ + fi”)m], where 1 and m are integers with m odd
and 1 > 0. It follows that L, = L( fi’+” + fi”nP’)mj.
3.47 (a) c = -i. (1~) c is an integer. (c) c = 0. (d) c is arbitrary. See the
answer to exercise 1.2.4-40 in [173] for more general results.
3.48 (Solution by Heinrich Rolletschek.) We can replace (a, (3) by ({ (3},
LX + \l3J ) without changing \na] + Ln(3]. Hence the condition a = {B} is
necessary. It is also sufficient: Let m = ]-fi] be the least element of the given
multiset, and let S be the multiset obtained from the given one by subtracting
mn from the nth smallest element, for all n. If a = {(3), consecutive elements
of S differ by either ci or 2, hence the multiset i.S = Spec(a) determines 01.
3.49 According to unpublished notes of William A. Veech, it is sufficient to
have a(3, (3, and 1 linearly independent over the rationals.