Concrete mathematics : a foundation for computer science

66 SUMS
34
35
36
Prove that if the sum tkeK ok is undefined according to (zsg), then it
is extremely flaky in the following sense: If A- and A+ are any given
real numbers, it’s possible to find a sequence of finite subsets F1 c Fl c
F3 (I ’ . . of K such that
IL ak 6 A - , when n is odd; t ak > A+, when n is even.
&Fn kEFn
Prove Goldbach’s theorem
1 = ;+;+;+:;+;+&+$+&+... = t’,
kEP k-’
where P is the set of “perfect powers” defined recursively as follows: Perfect power
corrupts perfectly.
P = {mn 1 m 3 2,n 3 2,m @ P}.
Solomon Golomb’s “self.-describing sequence” (f (1) , f (2)) f (3)) . . . ) is the
only nondecreasing sequence of positive integers with the property that
it contains exactly f(k) occurrences of k for each k. A few moments’
thought reveals that the sequence must begin as follows:
c+++x:i::::lk2
Let g(n) be the largest integer m such that f(m) = n. Show that
a s(n) = EC=, f(k).
b 9(9(n)) = Ed=, Wk).
c 9(9(9(n))) = ing(fl)(g(n) + 1) - i IL;:: g(k)(g(k) + 1).
Research problem
37 Will all the l/k by l/(k + 1) rectangles, for k 3 1, fit together inside a
1 by 1 square? (Recall that their areas sum to 1.1