Math 453: Elementary Number Theory, Spring 2008
Professor A.J. Hildebrand
Final Exam and Course Grades
Final Exam scores and course grades are now online and can be accessed as
usual. The maximal score on the final was 130 (5 points per subproblem),
the highest score was 123, the average 85, the median 91, and the 70th
and 30th percentiles 97 and 74, respectively.
Course grades were assigned on a linear scale, based on the overall
average, which in turn was computed using the weights of 1/3 final, 1/6
homework, and 1/2 midterms announced at the beginning of the class. The
cutoffs for letter grades were a bit more generous than those in place
after the midterms. In addition, I manually adjusted some of the grades
upwards to avoid hardships. As a result, nobody was within one
percentage point of the next higher grade; in fact, most gaps between
grades were 2 percentage points or more.
Final Exam Solutions are available under this
link.
Have a good break and enjoy your summer!
Course Policies, Exams, Grades
- Course Information Sheet.
The first-day handout. Everything you need to know about this class:
Syllabus, grading policies, office hours, etc.
-
Online Scores. Log in with your Net-ID and password.
The display shows the scores on all assignments and exams given out so
far. In addition, as of 4/17/08,
the display now also includes your current average score
and the corresponding curved letter grade. The Exam 2 grading information page explains
what all this means; check this page first before asking questions
about the display.
-
Midterm Exam 1: Wednesday, February 27.
-
Midterm Exam 2: Wednesday, April 16.
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Final Exam: Thursday, May 8, 8 am - 11 am.
Selected online resources ("Prime pages", etc.), and a list of alternate texts
that I have made available on library reserve.
Homework
- HW Assignment 1, due Monday, 1/28.
- HW Assignment 2, due Monday, 2/4.
- HW Assignment 3, due Monday, 2/11.
- HW Assignment 4, due Monday, 2/18.
- HW Assignment 5, due Friday, 2/22.
- HW Assignment 6, due Friday, 3/14.
- HW Assignment 7, due Monday, 3/31.
- HW Assignment 8, due Friday, 4/11.
- HW Assignment 9, due Monday, 4/28.
Class Notes
These are slightly updated and corrected versions of the sets of notes
distributed in class.
Class Diary
- Monday, 4/28:
- Lecture.
Wrapped up the continued fraction chapter with a review of approximation
properties of convergents, and an application to calendars.
Wednesday will be a Q and A session.
- Friday, 4/25:
- Lecture.
Continued fractions, continued: Approximation properties of
convergents.
- Read. Section 7.4.
- Wednesday, 4/23:
- Lecture.
Continued fractions, continued:
Convergents.
- Note on HW 9: Ignore the
last problem (Problem 40 in 7.5). One direction in this problem
requires a theorem from Section 7.6, which we haven't covered in class
and may not cover at all.
- Read. Section 7.3.
- Monday, 4/21:
- Lecture.
Continued fractions, continued:
The main theorems on the convergence of infinite c.f.'s,
the c.f. expansion of rationals, quadratic irrationals, and
nonquadratic irrationals. Examples of continued fraction expansions
of quadratic irrationals.
- Read. Sections 7.4, 7.5. (I will cover Section 7.3 on
Wednesday.)
- Friday, 4/18:
- Lecture.
Continued fractions, continued:
Basic definitions and concepts: finite and infinite
c.f.'s, simple c.f.'s, convergents, etc.
- Read. Section 7.2
- Wednesday, 4/16:
- Monday, 4/14:
- Lecture.
Continued fractions: Overview, motivation, and introductory
examples.
- Friday, 4/11:
- Lecture.
Question/Answer session for the midterm.
- Wednesday, 4/9:
- Lecture.
Integers, rationals, algebraic and transcendental numbers.
Irrationality proofs for squareroot of 2 and e.
(This is a prelude to Chapter 7, the final chapter in the
course; I will skip Chapter 6.)
- Monday, 4/7:
- Lecture.
Orders and primitive roots:
efficient computations.
- Friday, 4/4:
- Lecture.
Chapter 5, continued.
- Read. Sections 5.1 - 5.3
- Wednesday, 4/2:
- Lecture.
Started Chapter 5. Orders and
primitive roots.
- Monday, 3/31:
- Lecture.
The Quadratic Reciprocity Law. Applications.
- Read. Section 4.3
- Friday, 3/28:
- Lecture.
Wrapped up 4.2. Proof of Euler's Criterion and the multiplicativity of
the Legendre symbol
- Wednesday, 3/26:
- Lecture.
The Legendre symbol. Euler's Criterion. Applications.
- Read. Section 4.2
- Monday, 3/24:
- Lecture.
I started Chapter 4. Quadratic residues and nonresidues.
Motivation, definitions, and some basic results on residues and
nonresidues.
- Read. Section 4.1
- Friday, 3/14:
- Lecture.
Wrapped up Chapter 3, with a discussion of
perfect numbers, Mersenne primes, and the Euler-Euclid theorem
characterizing even perfect numbers.
I will start Chapter 4 after the break.
Enjoy the break!
- Read. Section 3.5 (Perfect numbers)
- Wednesday, 3/12:
- Lecture.
More applications the Dirichlet product.
The Moebius function and the Moebius Inversion Formula.
- Read. Section 3.6 (Moebius Inversion Formula).
- Monday, 3/10:
- Lecture.
The algebra of arithmetic functions; the Dirichlet product of arithmetic
functions, its properties and applications.
- Friday, 3/7:
- Lecture.
The number-of-divisors and sum-of-divisors functions.
- Read.
Sections 3.3 and 3.4
- Wednesday, 3/5:
- Lecture.
Notational conventions for divisor sums and products.
Multiplicative arithmetic functions.
- Read:
Section 3.1.
- Monday, 3/3:
- Lecture.
Some special values of the Euler phi function.
Connection with Fermat primes.
- Friday, 2/29:
- Lecture.
More on the Euler phi function. Multiplicativity of the phi function and
explicit formula. Carmichael's conjecture.
- Read:
Section 3.2.
- Wednesday, 2/27:
- Monday, 2/25:
- Lecture:
An application of the gcd and the Euler phi function:
Counting lattice points visible from the origin.
- Friday, 2/22:
- Lecture:
Computational aspects: Comparison
of different algorithms for the gcd, and the Wilson and Fermat primality
tests.
- Wednesday, 2/20:
- Lecture:
Reduced systems of residues.
Proof of Euler's Theorem.
- Monday, 2/18:
- Lecture:
More on Wilson's and Fermat's Theorems.
Applications of Fermat's Theorem.
Pseudoprimes. Euler's Generalization of Fermat's Theorem.
- Read:
Section 2.6.
- Friday, 2/15:
- Lecture:
Wilson's Theorem, Fermat's Little Theorem.
Pseudoprimes.
- Read:
Sections 2.4/2.5.
- Wednesday, 2/13:
- Lecture:
The Chinese Remainder Theorem.
- Read:
Section 2.3.
- Monday, 2/11:
- Lecture:
Linear congruences in one variable. Existence of a solution, and
an algorithm to construct a solutions.
- Read:
Section 2.2.
- Friday, 2/8:
- Lecture:
Wrapped up Section 2.1 with a discussion of equivalence relations,
residue classes, complete residue systems. Started Section 2.2 with an
overview, and a formulation of the basic problem of solving a linear
congruence.
- Wednesday, 2/6:
- Lecture:
Congruence magic, continued. Applications to Mersenne and Fermat
numbers, divisibility tests.
- Read:
Read Section 2.1 if you have not done so. Also, review the concepts of
an equivalence relation and equivalence classes in Appendix B of
Strayer. I may spend a bit of time on Friday on this, but it would be
good if you have at least some familiarity with these concepts.
- Notes on HW 3:
- Problem 22(a)(b)(c):
This problem asks for divisibility criteria for "repunits".
You can use (without proof) any of the divisibility tests given in
Section 2.1 and in Problems 18 and 19 (Problem 18 deals with
divisibility by 11, and is part of the assignment. Problem 19 gives a
test for 13.)
- Non-turn-in problems. Through a botched cut-and-paste job,
some of the non-turn-in problems from the last assignment were
mistakenly included. The affected problems are the last five on the
list: 11-15. Just ignore those! (The asterisk problems are all correct,
as are the first four non-asterisk problems (7-10).)
- Monday, 2/4:
- Lecture:
Started Chapter 2.
Definition and elementary properties of congruences.
- Read:
Section 2.1
- Friday, 2/1:
- Lecture:
Wrapped up Chapter 1. More applications of the Fundamental Theorem of
Arithmetic. Characterization of divisibility, gcd, and lcm, in terms of
prime factorizations.
- Read:
Section 1.5 (Fundamental Theorem of Arithmetic).
- Wednesday, 1/30:
- Lecture:
Least common multiples, continued. Started the discussion of the
Fundamental Theorem of Arithmetic and its applications.
- Read:
Section 1.5 (Fundamental Theorem of Arithmetic).
- Monday, 1/28:
- Lecture:
The Euclidean algorithm. Least common multiples.
- Read:
Section 1.4, which covers the Euclidean algorithm, and the final part
of Section 1.5 (beginning with p. 28). The first part of 1.5 is devoted
to the Fundamental Theorem of Arithmetic which we'll get to on
Wednesday.)
- Friday, 1/25:
- Lecture:
Gcd, continued: Characterization of gcd in terms of
linear combinations of integers. Application to proof of Euclid's
Lemma. More about primes: Prime Number Theorem, Mersenne and Fermat
primes.
- Read:
I have not gotten beyond Section 1.3 in the lectures, so there is
no new reading assignment. However, if you have not done so, make
sure to carefully read Sections 1.1 - 1.3. The lectures and the text
complement, rather than duplicate, each other, you are expected to be
familiar with material covered in the corresponding sections of Strayer,
both for the in-class exams and for the hw assignments, including topics
(such as the sieve of Eratosthees) that I did not cover in detail in the
lectures.
- Wednesday, 1/23:
- Lecture:
Greatest common divisors (gcd). Definition, examples, basic properties.
The gcd and linear combinations of integers. Application to McNugget,
postage stamp, coin changing type problems.
- Read: Strayer, Section 1.3.
- Friday, 1/18:
- Lecture:
Primes and composite numbers. Euclid's Theorem (on the infinitude of
primes).
- Read: Strayer, Section 1.2.
- Wednesday, 1/16:
- Lecture:
Divisibility: definition and basic properties. Divisibility in more
general settings. The division algorithm.
- Read: Strayer, Sections 1.1 and 1.2.
- Monday, 1/14:
- Lecture:
Course Overview.
Last modified: Fri 09 May 2008 09:47:05 AM CDT
A.J. Hildebrand