THEORY OF NUMBERS, Math 115 A Homework 7 Due Wednesday November 27 2002

1. (not from the book) Prove or disprove the following statement: For every odd number n, σ(n) ≤ 2n − 1. HINT: Try to experiment with LOTS of numbers, use your intuition.

2. (7.1, 8) Show that there is no positive integer n such that ϕ(n) = 14.

3. (7.1, 18) Show that if m, k are positive integers then ϕ(m^{k}) = m^{k−1}ϕ(m).

4. (7.2, 12) Show that the equation σ(n) = k has at most a finite number of solutions when k is a positive integer.

5. (7.2, 11) What is the product of the positive divisors of a positive integer? Write a formula for it.

6. (7.3, 4) Find a factor of each of the following integers: 2^{1}11 − 1, 2^{289} − 1, and 2^{46189} − 1.

7. (7.4, 6) Find all composite numbers n between 100 and 200 with μ(n) = −1.

8. (7.4, 8) Find the value of the Mertens function M(n) for n = 100.

9. (7.4, 22) Let n be a positive integer. Show that Π_{d|n}μ(d) equals (1) -1 if n is a prime, (2) Zero if n has a square factor, and (3) 1 if n is square-free and composite.