Topics Covered in Math104b Winter 2009
1. Primitive roots
2. Characters
3. Gauss sums
4. Quadratic reciprocity
5. Continued fractions
6. Pell equation
s
7. Transcendental numbers and Liouville theorem
s
8. The transcendence of e.
9. Distribution of primes
1
Characters.
Let G be a commutative group with identity element e. Then we recall that
a character is a group homomorphism
:G!C =C
f0g:
b
That is (ab) = (a) (b) for a; b 2 G. We denote by G the set of characters
of G. The following properties are satised
Practice Problems for the Midterm
1, Let m 2 Z and m > 1: Show that if a 2 Z and gcd(a; m) = gcd(a
1; m) = 1 then
1 + a + a2 + : + a (m) 1 0 mod m:
(Here (m) is Euler -function, as usual.)
s
2. Let p be a prime. Recall that the principal Dirichlet charact
Extra Exercises for 2/2/09
The rst part of this assignment is to install Pari gp in your computer. The
references to pari are to the program gp.exe that will be available when the
program is installed.
p
1. Calculate the continued fraction expansion of 3
Extra Problems 3/4/09
1. Clearly x2
for 1 x 40.
x + 41 is not a prime for x = 41: Show that it is a prime
2n
2 2n
for n 4. Use this lower bound in place of
2n
n
the one we used (2n ) to get a better lower bound for (x).
2. Show that
3. Let pn be the nth p
Extra Problems 3/3/09
1. Prove that (m)
m
3
for m 2 Z, m
33:
2. Let (x) be the number of integers 1 n
prove that
(x)
= 0:
lim
x1+1 (x)
x such that n is a square
pn
3. Let pn be the nth prime (in pari gp pn = prime(n). Calculate n log(n)
for n = k (1000);
Extra Problems for 2/17
1. Show that if m > 0, m 2 Z then
p
m + m2 + 4
= [m; m; m; m; :]:
2
2. Show that if m > 0 and m; a 2 Z then
[a; m; m; m; :] =
2a
p
m + m2 + 4
:
2
3. Prove that if is irrational with > 1 then if has simple continued
fraction [q0 ; q
Extra Problems for 1/9/09
uv
= 1 . Show that if d
w1
p
is a positive integer that is not a square of an integer and d = [q0 ; q1 ; q2 ; :]
is its simple continued fraction then
p
u d+v
p
= [v; w; q0 ; q1 ; :]:
w d+1
p
p
7+2
2. Calculate the simple continu
Extra Exercises for 1/26
1. Show that if p is an odd prime and a 2 Z is such that p - a and k
then
k1
ap (p 1) 1 mod pk :
1
2. Prove under the hyptheses in 1. that
k 1 (p
ap
1)
ap
1
pk
1
p
1
mod p:
3. Let = e2 i=3 : Find f (t) = a0 + a1 t + a2 t2 + t3 wit
b
Let G be a commutative group. We dene G to be the set of all group
homomorphisms from G as a group under addition to C = C f0g. Such
homomorphisms are called characters. We have dened a group structure on
b
b
G as follows: If 1 ; 2 2 G then
(
1 2 )(x)
=