MATH 3391
Course Outline
Math 3391 Elementary Number Theory
Fall Semester 2004
Time: 9:00 – 9:50 ASB 332
Instructor: Max Warshauer
Website: www.txstate.edu/mathworks
E-mail: max@txstate.edu
Office: Academic Support Building (ASB) 110 Texas
Mathworks
Phone: 245-3439 (office) 396-8281 (home)
Office Hours: 10-11 MWF Other times
by appointment
RequiredText: None
Homework
Policy: Homework will be assigned every class period.
Students are encouraged to work together on the assignments,
and then write up their own individual solutions. Written
assignments will be turned in the following class. Redone
problems will be accepted if resubmitted by the next
class after the assignment is returned. All assignments
should be kept together in a homework notebook.
Bonus: Students may write
a short article for Math Explorer Magazine
(grades 4-8). The form of the article is open-ended.
This can add up to 4 points to the final average.
Make-up Policy: No make-up tests will be given
without prior arrangement.
Attendance Policy: Attendance is encouraged but
not required. I hope that you will all enjoy class
and want to come every day.Students will be required
to make class presentations about their homework problems
which will become part of the homework grade.
Final Average:[(Midterm) + (Final) + 2 (Homework)]/4
+ Bonus
Drop Policy: The final date to withdraw with
NO RECORD assigned is Sept. 10. If you drop after Sept.
14, your grade will be determined using the formula
above. This may result in an “F” in the
course. If you drop the course, it is important to stop
by and see me to verify whether you will receive a “W”
or “F”. The final date to drop is November
22.
Disability Needs: Students with special needs,
as documented by the Office of Disability Services,
should identify themselves at the beginning of the semester.
We will be happy to work with any students with special
needs.
Academic
Honesty Policy: We follow the Texas State Academic Honesty
policy, as outlined in UPPS No. 07.10.01.
Course Description:
Elementary Number Theory is ideally suited for the honors
program because students at different levels of mathematical
maturity can all participate in and learn from such
a course. Students begin by studying simple ideas about
the integers, where they already have a well-developed
intuition. To paraphrase David Gries (The Science of
Programming), one should never take basic principles
for granted, for it is only through careful application
of simple fundamental ideas that progress is made. The
division algorithm is studied in detail, and is seen
to have far-reaching consequences throughout the course.
Done repeatedly, it yields Euclid’s algorithm
and the solution to linear Diophantine equations. Properties
of divisibility also lead naturally to modular arithmetic
and related questions about quadratic forms. The students
explore quadratic residues, culminating with Legendre
Symbols and a development of the Law of Quadratic Reciprocity.
An outline of topics which may be covered includes:
Bases and modular arithmetic Euclid’s algorithm
Foundations and axioms Unique factorization for Z
Methods of proof (direct, indirect) Diophantine equations
Induction, well-ordering Equivalence relations
Division algorithm Congruences
Complete, reduced residue systems Pseudoprime test,
strong pseu. test
Chinese remainder theorem Public key encryption
Polynomial congruences Arithmetic functions
Wilson’s theorem, Euler’s theorem Recurrence
relations
Iterative and recursive algorithms Moebius inversion
formula
Continued fractions Gaussian integers, units
Fractional expansions for rationals Primes in Z[i]
Quadratic surds Quadratic residues, Legendre symbol
Approximation theorems Quadratic reciprocity
Pell’s Equation Summary and discussion
Topics will be adjusted to the needs of the class.
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