Mathematics 400c
Homework (due Mar. 3)
A. Hulpke
31)
Let µ be the Möbius function. Show that for all positive integers n:
(
1 n=1
∑ µ(d) = 0 n > 1
d|n
32)
For positive integers n and k we define σk (n) = ∑ d k . Show that σk (n) is multiplicative.
d|n
33)
Let f and F be multiplicative number theoretic functions, such that f (x) = ∑ µ(d)F(n/d).
Show that F(n) = ∑ f (d). (This is the converse of Möbius’ theorem.)
d|n
d|n
34)
The Mangoldt function is defined as
(
ln p if p is the only prime factor of n,
∆(n) =
0
if n 6= pm .
Show that its summatory function is ln(n).
35) Let f (x) ∈ Z[x] be a monic polynomial (i.e. the leading coefficient is 1). Let p be a
prime number.
a) Show that if f (a) ≡ 0 (mod p) then there exists a monic polynomial g(x) ∈ Z[x] such that
f (x) ≡ g(x) · (x − a).
(Hint: Use division with remainder.)
b) Using the fact that polynomials modulo p have a unique factorization, conclude that a
polynomial of degree n has at most n different roots modulo p. (This is sometimes called
“Lagrange’s Theorem”.)
36∗ ) a) Let n be a Carmichael number and p|n. Show that (p − 1)|(n − 1).
b) Let n = p1 p2 · · · pk be a product of distinct primes (such a number is called “squarefree”)
such that pi − 1|n − 1. Show that n is a Carmichael number.
c) In fact one can show (proof later):
A squarefree integer n is a Carmichael number if and only if (p − 1)|(n − 1) for
every p|n.
Using this criterion, find a Carmichael number > 2000 (you may use GAP or another
compurer algebra system).
Problems marked with a ∗ are bonus problems for extra credit.