SUMMER 2012 Assignment #2

Due May 30 at the beginning of class.

1. Prove that for any non-negative number x, 
   floor(x) = SUM( round(x/2^k), k >= 1 ). 

2. Prove that if 0 < alpha < 1 is irrational, then for all positive
   integers n, {n*alpha}+{n*(1-alpha)} = 1.

3. Write a Maple and a Sage routine to compute the following sum
   a(m) = SUM( floor((m+k)/2^k) PHI(k) ), k=0..m-1 ), where PHI(k)
   is the Euler totient function.    Compute a(m) for m = 1..16.

4. On the basis of the number that you see in problem 3, make a 
   conjecture about the value of a(m) and try to prove it. 
   NOTE: This is not an easy problem.

5. Prove the "Note:" in the solution to exercise 3.21 on page 508.

6. Find three irrational numbers a,b,c such that 1/a + 1/b + 1/c = 1.
   Compute the spectrum of each up to n <= 32 using Maple or Sage
   and show the output.