Mathematics 115
Fall, 2012
TuTh
12:40-2PM,
4 Evans Hall
Overview
This course is elementary in the sense that no
specific course
is needed as a prerequisite.
On the other hand,
the ability
to read and write proofs is pretty much essential.
Math 55 and Math 113 provide helpful background.
As we encounter objects that can be regarded as groups, rings,
fields, homomorphisms,..., I will mention the connection
to Math 113 in class.
Textbook
An Introduction to the
Theory of Numbers, Fifth Edition
by
Ivan
Niven, H. S. Zuckerman and
Hugh L. Montomery.
Although the current edition
was published 20 years ago, this book remains one of the
definitive introductions to the subject. It is renowned for its
interesting, and sometimes challenging, problems.
Please see
Montgomery's
home page for the book and especially his lists of typos and errors
in the book. Note that there are multiple lists because the book has
been reprinted several times.
This book is not cheap, but it should be easy to find used copies:
“Niven & Zuckerman” (as the book is widely known) has been used
repeatedly at Berkeley.
Chapter-by-Chapter Description
- Chapter 1: Divisibility, the Euclidean algorithm,
primes and the Fundamental Theorem of Arithmetic, proofs of the
infinitude of primes, the binomial theorem.
Much of this material will be familiar from Math 55, but I
hope that the book and my lectures will provide additional perspective.
We can also do some computations with sage
(see below).
- Chapter 2: Congruences, Euler's phi function,
the ring of integers mod m, Fermat's little theorem
and its generalization by Euler,
Wilson's theorem, Fermat's theorem characterizing integers that
are sums of two squares,
polynomial equations mod m, the Chinese remainder theorem,
Pollard's rho method for factoring,
RSA cryptography, Hensel's lemma, primitive roots, quadratic residues
and higher residues.
This chapter ends with some material on groups, rings and fields.
We will not discuss this material in any depth in class, but I will
allude to it from time to time, as I explained in connection with
Math 113.
- Chapter 3: Quadratic reciprocity, the Jacobi symbol and
applications, binary quadratic forms.
I hope to give several proofs of quadratic reciprocity, starting
with the proof in the book. This means, in particular, that I will
be lecturing on material not in the book.
Similarly, when speaking about binary quadratic forms, I hope to
explain a new way of looking at the classical results
that was discovered recently by
Manjul Bhargava.
- Chapter 4: Some functions of number theory.
We will discuss only some of the material in this chapter.
For example, we will prove the Moebius inversion formula.
- Chapter 7: We will discuss continued fractions, as time
allows.
For example, we will see in the beginning of the chapter that
the Euclidean algorithm of chapter 1 is a method for finding the
continued fraction expansion of a rational number.
Sage
Sage is a
free open-source mathematics software system
that does
number theory calculations that will illustrate
and
illuminate the material of the course.
Even before the semester begins, you can
become familiar with sage by taking
the tour
and then experimenting with the software.
When I taught Math 116 last semester,
I projected a sage notebook from my laptop for a substantial
fraction of each lecture period. Don't be surprised if I
continue in that direction this semester.
I will try to assign interesting homework problems that require
sage for calculations.
You can
download
the software for your Windows, Linux or MacOS X box.
Alternatively, you can run sage online at
http://www.sagenb.org/
after you create an account for yourself.
Examinations
- First
midterm exam, Thursday, September 20, 2012, in class
(questions and skeletal solutions, mean 19.14, standard deviation 7.29).
- Last
midterm exam, Thursday, October 25, 2012, in class
(questions and skeletal solutions, mean 19.79, standard
deviation 7.55).
- Final
examination, Friday, December 14, 2012, 8-11AM in
3113
Etcheverry Hall
(questions and skeletal solutions).
On
this
54-point exam, the mean was 34.26 while the standard deviation was 11.84.
Please do not plan travel on the dates of these exams.
If you believe that you have a conflicting obligation because of an
intercollegiate sport or other
extracurricular activity,
please read
these
guidelines
immediately.
For
practice exams, you might consult the web pages for my previous
Math 115 courses
and for
a
recent course by
Martin Olsson.
You may also consult
Richard Borcherds's
Math 115
page for Fall, 2003.
Grading
Course grades
will be based on a composite numerical score
that is intended to weight
the course components roughly as follows:
midterm exams 15% each, homework
25%, final exam 45%.
When I taught this course in 2011, there were 33
registered students. Grades were distributed as follows: 11 As,
14 Bs, 4 Cs, 4 D/F. This rough distribution ignores +'s and -'s.
Some students took the course P/NP. Their letter grades
were
converted to P or NP when I entered the final grades.
For this course (Fall, 2012), there were 36 registered students. They received 17 As, 13 Bs, 3Cs and 3Fs.
The students with the a P/NP grading option had their grades converted when final grades were entered.
You can look at the
course evaluations that my Math 115
students wrote in December, 2011 as well as the
evaluations for this course.
I encourage suggestions and comments about my teaching style and the
evolution of the course. You can make them anonymously in various ways
(e.g., by slipping a note under my office door) or just present them
in email or a face-to-face conversation.
Homework
- Assignment due August 30, 2012:
§1.2, problems 1, 2 and 3: all parts, using sage;
also problems 4b, 5, 7, 15, 25, 27, 28, 47
- Assignment due September 6, 2012:
§1.3, problems 2, 8, 10, 11, 13, 16, 17, 26, 28, 31
- Assignment due September 13, 2012:
- §1.3, problems 42, 44, 48
(for the last problem, see the first lines
of the
basic properties section of the
wikipedia Fermat
number entry)
- §1.4, problems 3, 4
- §2.1, problems 6, 13, 26, 30 (check using sage), 34, 35, 36, 37, 43
- Assignment due September 20, 2012:
- §1.2, problem 50
- §1.3, problems 27, 29, 36
- §1.4: Let n = 5k + j with k at least 1 and j = 0, 1, 2, 3 or 4.
Show that k is congruent mod 5 to the binomial coefficient "n choose 5".
- §2.1, problems 33, 40
- §2.2, problems 8, 9
- Assignment due September 29, 2012:
§2.3, problems 4, 8, 13, 14, 17, 18, 26, 27, 29, 30, 39
- Assignment due October 4, 2012:
- §2.7, problems 1 (using sage if possible), 6, 12
- §2.8, problems 3, 12, 16, 18, 20, 23
- Assignment due October 11, 2012:
- §2.8, problems 24, 25, 29, 30, 31
- §2.9, problems 1ad, 7
- §3.1, problems 4 (just use sage), 5 (use sage to compute), 6 (use sage to avoid tedium)
- Assignment due October 18, 2012:
- §3.1, problems 13, 14, 15, 16, 17, 18
- §3.2, problems 6, 7, 8, 11, 13
- Assignment due October 25, 2012:
- §2.3, problem 37
- §2.7, problems 13, 14
- §2.8, problems 33, 34, 35
- §3.3, problems 14, 15
- Assignment due November 1, 2012:
- §4.1, problems 2, 5, 9, 14, 17, 19, 34
- §4.2, problems 10, 12, 16, 19
- Assignment due November 8, 2012
- Assignment due November 15, 2012:
- §10.1, problems 3, 4
- §10.2, problems 1, 2, 4, 5, 6
- §10.3, problem 3
- §10,4, problem 2
- Assignment due November 27, 2012:
- §5.4, problems 14
- §5.6, problems 1, 9, 10 (skipping the part about "nonsingular")
- §5.7, problems 9, 10
- Assignment due December 6, 2012:
- §2.4, problems 16, 17, 18
- §5.8, problem 4
- Find the order of the point P = (94269158776925 , 1102841572571055)
on the elliptic curve
defined by y^2 = x^3 + 183821385707290x + 1153449657807210
over the ring of integers modulo M=1636998688431221.
Do this by using the elliptic curve factoring method to factor
M; then use sage to compute the order of P
modulo each of the factors of M.
Some web resources related to the course
- What is the smallest
prime? by
Chris K. Caldwell
and Yeng Xiong.
- Reinventing the Classroom
by Harry R. Lewis.
Should we organize Math 115 this way?
- The on-line encyclopedia of integer
sequences
- The prime pages
- Eckford Cohen's article
about the prime factorization of n!, which includes a weird proof that
there are infinitely many primes. (You have access to the
full
article when authenticated as coming from Berkeley.)
- A
compendium
of proofs of Fermat's 1654 theorem to the effect that primes that are 1 mod 4
are sums of two squares.
See especially
Don Zagier's
proof
of the theorem.
See also the PlanetMath
account
of the proof, which includes a simple proof of the uniqueness.
- While we're at it, you might enjoy Don Zagier's article
The
first 50 million prime numbers
- A fairly old list of
alternative
references for Math 115.
- An interesting and fairly recent
discussion
about the question of whether ((p-1)/2)! is +1 or -1 mod p when p is 3 mod 4.
This residue class is 1 for p = 3, 23, 31, 59, 71 and 83 but -1 when p is
7, 11, 19, 43, 47, 67, 79. It's hard to see a pattern, isn't it?
- An alternative proof of the law of quadratic reciprocity.
If you prefer, you can consult the
original
1872 manuscript by Zolotarev.
In fact, you may find this
mathoverflow
discussion to be of great interest.
One manuscript that mathoverflow
refers to is this
shortened
proof by
Wouter Castryck.
- Just to see if anyone is reading these links, here's an
article
about a lecture that I gave on October 16, 2011 at
Humboldt State University.
I gave two more lectures the next day.
- Wikipedia's
discussion
of the Lucas-Lehmer test.
This web page was the basis for my lecture on November 3, 2011.
- William Stein's book
Elementary
Number Theory: Primes, Congruences, and Secrets.
- Henry
Cohen's article
A Short Proof
of the Simple Continued Fraction Expansion of e.
- Hendrik Lenstra's
2002 article
on Pell's equation.
See also
The
Number of Real Quadratic Fields Having Units of Negative Norm
by
Peter Stevenhagen.
Calendar
The calendar that follows (at least if you're logged into gmail!)
attempts to call your attention to "events" of interest to math 115 students:
class meetings, office hours, exams, optional class
get-togethers (coffee, lunch, breakfast), and special lectures for
undergraduates.
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