Mathematics 124
MIDTERM
Collaboration: On the midterm no collaboration is allowed. Show all your working, and
write up your solutions as neatly as possible. The points on the Exam total 105, although
only the rst 100 points will count. If you have any que
Mathematics 124
HOMEWORK ASSIGNMENT # 3
DUE, Friday, October 17
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share
Mathematics 124
HOMEWORK ASSIGNMENT # 1
DUE, Friday, October 3
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share w
Mathematics 124
HOMEWORK ASSIGNMENT # 4
DUE, Friday, October 24
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share
Mathematics 124
HOMEWORK ASSIGNMENT # 5
DUE, Friday, October 31
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share
Math 124 Homework 4 Solutions
1. Consider the coecient of xp in the expansion of (1 + x)2p = (1 + x)p (1 +
2p
x)p . On the left side, the coecient of xp is
. On the right side, it is
p
given by
p
p
2
p
p
p
=
i
pi
i
i=0
i=0
p
i
Recall that for i = 0, p,
is
Math 124 Homework 2 Solutions
1. First, cancel out the factors of p so that we avoid division by zero:
2p
p
2p(2p 1)(2p 2) (1)
p2 (p 1)2 (p 2)2 (1)2
=
=
2p(2p 1)(2p 2) (p + 1)
p(p 1)(p 2) (1)
=
2(2p 1)(2p 2) (p + 1)
(p 1)(p 2) (1)
Now, each factor in the
1
Lecture 1
Number theory is generally split into two elds: Algebraic number theory: Sol.
to poly. eqs over Q, Z, /n (Book for this part: Davenport) Analytic number
theory: Asymptotic estimates on distributions of various classes of numbers.
(Dirichtlet t
Math 124 Homework 5 Solutions
1. 163 12 (mod 2) gives p = 2 the smallest prime.
2a. First, consider q = 2. We know 2 is not a quadratic residue if and only
if p 3, 5 (mod 8). By Dirichlets theorem, there are innitely many primes
p of the form 3 + 8n, so t
Mathematics 124
HOMEWORK ASSIGNMENT # 2
DUE, Friday, October 10
Collaboration: On the homework sets, collaboration is not only allowed but encouraged.
However, you must write up and understand your own individual homework solutions, and
you may not share