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Jordan University of Science and Technology
Faculty of Science
Department of Mathematics and Statistics
Semester 2006/2007
Course Syllabus For Math. 345 (Number Theory)
Course Information
Course Title
Number Theory
Course Code
Math. 345
Prerequisites
---
Course Website
Instructor
Dr. Ibrahim Al-Ayyoub
Office Location
Ph2 Level 0
Office Phone #
23451
Office Hours
Sun. Tue. Thu. 3 - 4
E-mail
[email protected]
Teaching
Assistant(s)
See the list at the department
and Mon. Wed. 2 - 3
Course Description
Schedule Spring Semester 2007
Weeks
1
2
3
Weeks of
4
5
March 4
March 11
Feb. 11
Feb. 18
Feb. 25
Section and suggested exercises
1. Divisibility
1.1 The GCD and LCM. 18, 21, 22, 23, 30, 31, 34, 36.
1.2 The Division Algorithm. 7, 9, 17-24, 28, 29, 30, 35, 36.
1.3 The Euclidean Algorithm. Odd 1-9, 23, 24, 25, 28.
1.4 Linear Combinations. Odd 1-19, 20, 24, 32, 36, 37, 39.
1.5 Congruences Odd 1-15, 27, 30, 39, 40, 44, 46, 47.
1.6 Mathematical Induction. 15, 16, 19, 21, 29, 31, 35, 43, 52, 61-64.
2. Prime numbers
2.1 Prime factorization. 5, 15, 17, 19, 20, 24, 25, 27, 30, 32, 39.
2.2 The Fundamental Theorem of Arithmetic. 5, 9, 19, 22,
24, 25, 26.
2.3 The Importance of Unique Factorization
1, 3, 11, 13,
21, 22, 35, 37, 42.
6
7
8
March 18
March 25
April 1
2.4
2.5
2.6
3.
3.1
3.2
Prime Power Factorization. 37, 43, 46, 52.
The Set of Primes in Infinite. 5, 7, 11, 12, 14, 16, 19.
A Formula for τ(n). 1, 5, 7, 16, 18, 20, 23, 35, 40, 41, 43, 45
Numerical Functions
The Sum of the Divisors. 13, 38, 40, 41, 44.
Multiplicative Functions. 1, 3, 5, 7, 8, 10, 14, 15, 17, 18, 19,
22, 27, 28.
3.3 Perfect Numbers. 1, 4, 19, 20, 21, 26, 30, 31.
3.4 Mersenne and Fermat’s Numbers. 6-13, 14, 15, 18, 19, 20, 21.
3.5 The Euler Phi Functions. 11, 12, 17, 18, 26, 28, 31, 33, 34, 37,
40, 41, 42, 48, 50.
9
10
11
April 8
April 15
April 22
3.6 The Möbus Inverson Formula. 1, 3, 6, 8, 9, 12, 15, 17, 23.
4. The Algebra of Congruence Classes
4.1 Solving Linear Congrunces. Odd 25-35, 42-48, 51.
4.2 The Chinese Remainder Theorem. Odd 1-21, 31-36, 44, 45, 46,
47, 51.
4.3 The Theorems of Fermats and Euler.
Odd 1-19, 25, 28,
31, 34, 35, 37, 40, 43, 47, 49.
12
13
14
April 29
May 6
May 13
4.4 Primality Testing. 7, 9, 14, 17, 19, 20.
4.5 Public-Key Cryptography. Odd 1-13, 27, 29.
7. Diophantine Equations
7.1 Pythagorean Triples. Odd 1-11, Odd 21-31, 34
7.2 Sums of two squares. Odd 1-9, 17, 19, 20, 21, 22, 27, 28.
7.3 Sums of four squares. Odd 1-17, 22, 23, 26, 27, 28
7.4 A Special Case of Fermat’s Last Theorem. 19, 20.
15
May 20
16
May 27
Title
Author(s)
Publisher
Year
Review
Final Exam (TBA)
Textbook
Elementary Number Theory
Charles V. Eynden,
McGraw-Hill
Edition
2001
Second Edition
Book Website
Other references
Some handouts that will be given at class.
Assessment
Assessment
Date
Percentage
First Exam
2007- 3 -15
20%
Second Exam
2007- 4 -15
20%
Third Exam
2007- 5 -13
20%
Final Exam
Be announced by university
There are suggested exercises
foe each section in the above
schedule (not to be
collected, or graded by the
instructor)
It is expected that the
student to take a part in the
discussion that usually goes
on every class. Also, the
student will take a part in
writing proofs. It is very
crucial to work all the
suggested exercises and write
down the proof
professionally. Discussion on
these exercises with the
fellow students is very
constructive. Some homework
sessions will be held through
the semester. Questions may
be posed to the instructor at
class or at the office hours.
Attendance is required by the
40%
Assignments
Participation
Attendance
university rules. If for
reason a student reaches
15% warning limit, he or
will be prohibited from
participating in the
subsequence exams and
receives a grade of “35”
the course.
some
the
she
in
Course Objectives
To have a feel of the beauty and the challenge of
the theory of numbers.
Learn how to use the Euclidean algorithms for
solving some of Diophantine equations.
Understand the concept of the Fundamental Theorem
of
Arithmetic
and
the
importance
of
unique
factorization.
Percentage
10%
15%
15%
Dealing with some numerical function, such as, the 25%
Euler Phi function, and their applications.
Working with systems of linear congruence and learn 15%
Chinese Remainder Theorem as well as the Fermat’s
and Euler theorems and their applications.
Learn the intuitive approach of the improper 10%
integrals and techniques to evaluate such integrals.
To prove a special case of Fermat's Last Theorem.
10%
Teaching & Learning Methods
General Learning Objectives
1. Understand the concept of GCD and LCM and use them to solve
Congruence equations.
2. Learn the importance of uniqueness of factorization of
integers.
3. Learn some numerical functions and their applications.
4. Deal with Diophantine equations and Fermat Last Theorem.
Learning Outcomes: Upon successful completion of this course,
students will be able to
Related
Will be able to
Reference(s)
Objective(s)
Divisibility
Do some analysis and computations
related to the GCD and the LCM.
Apply the division and the
Euclidean algorithms to solve some
linear combination as well as some
Chapter 1
linear congruencies.
Apply principle of mathematical
induction I to some number
theoretic problems.
Prime numbers
Find the prime factorization of an Chapter 2
integer.
Prove some basic facts about the
prime integers.
Use the uniqueness of prime
factorization to redefine the LCM
and the GCD
Construct a formula for the number
of divisors of and integer, that
is, for τ(n).
Numerical
Functions
Construct a formula for the sun of Chapter 3
the divisors of and integer, that
is, for σ(n).
Decide whether a Mersenne of
Fermat number is prime.
Characterize the even perfect
numbers.
Prove basic facts about
multiplicative inverse.
The Algebra
Congruence
Classes
of
Apply
the
Chinese
remainder Chapter 4
theorem to solve system of linear
congruencies.
Apply some primality testing.
Diophantine
Equations
Perform ciphering and de-ciphering
in Cryptography.
Construct and characterize
Chapter 7
Pythagorean triples
Identify the integers that can be
written as a sum of two, three, or
four squares.
Understand a proof of a special
case of Fermat’s last theorem.
Useful Resources
1. David M. Burton, Elementary Number Theory, McGraw-Hill, 6th Ed,
2005.
2. Rosen, Kenneth H. , Elementary number theory and its
applications, Mass.:Addison-Wesley,1988.
3. Edwards, Harold M. , Fermat's Last Theorem; A Genetic
Introduction to Algebraic Number Theory, Graduate Texts in
Mathematics , Vol. 50, 2000.
4. Koblitz, Neal, A Course in Number Theory and Cryptography,
Graduate Texts in Mathematics , Vol. 114, 1994.
5. Cohen, Henri, A Course in Computational Algebraic Number Theory,
Graduate Texts in Mathematics, Vol. 138, 2000.
Additional Notes
A word of advice
The nature of the topic of Number Theory is very technical, that
is, it depends on lot of definitions and techniques. Therefore, in
order to understand the topic very well you need to keep it up with
the lectures by working all the examples and the suggested
exercises. This is very important in order to do well in the
course.
Note Some mathematical software to do some applications on the
course, such as, Mathematica or Maple.