Concrete mathematics : a foundation for computer science

96 INTEGER FUNCTIONS
Basics
10 Show that the expression
is always either 1x1 or [xl. In what circumstances does each case arise?
11 Give details of the proof alluded to in the text, that the open interval
(a.. (3) contains exactly [(31 - [a] - 1 integers when a < l3. Why does
the case a = (3 have to be excluded in order to make the proof correct?
12 Prove that
n
H L n+m-1
- =
m m J ’
for all integers n and all positive integers m. [This identity gives us
another way to convert ceilings to floors and vice versa, instead of using
the reflective law (3.4).]
13 Let a and fi be positive real numbers. Prove that Spec(a) and Spec( 6)
partition the positive integers if and only if a and (3 are irrational and
l/a+l/P =l.
14 Prove or disprove:
(xmodny)mody = xmody, integer n.
15 Is there an identity analogous to (3.26) that uses ceilings instead of floors?
16 Prove that n mod 2 = (1 - (-1)“) /2. Find and prove a similar expression
for n mod 3 in the form a + bw” + CW~“, where w is the complex number
(-1 +i&)/2. Hint: cu3 = 1 and 1 +w+w’=O.
17 Evaluate the sum &Gk 1 such
that
for all real x 3 1.