Concrete mathematics : a foundation for computer science

Research problems
3 EXERCISES 101
49 Find a necessary and sufficient condition on the nonnegative real numbers
a and p such that we can determine a and /3 from the infinite multiset
of values
59 bet x be a real number 3 @ = i (1 + &). The solution to the recurrence
Zo(x) = x7
Z,(x) = Z,&x)'-1 , for n > 0,
can be written Z,(x) = [f(x)2”1, if x is an integer, where
f(x) = $nmZn(x)1'2n ,
because Z,(x) - 1 < f (x)2” < Z,(x). What interesting properties does
this function f(x) have?
51 Given nonnegative real numbers o( and (3, let
Sw(a;P) = {la+PJ,l2a+P1,13a+P1,...}
be a multiset that generalizes Spec(a) = Spec(a; 0). Prove or disprove:
If the m 3 3 multisets Spec(a1; PI), Spec(a2; /32), . . . , Spec(a,; &,,)
partition the positive integers, and if the parameters a1 < a2 < ’ . . < a,,,
are rational, then
2m-1
ak = - 2k-1 ’ for 1 6 k < m.
52 Fibonacci’s algorithm (exercise 9) is “greedy” in the sense that it chooses
the least conceivable q at every step. A more complicated algorithm is
known by which every fraction m/n with n odd can be represented as a
sum of distinct unit fractions 1 /qj + . + . + 1 /qk with odd denominators.
Does the greedy algorithm for such a representation always terminate?