COURSE DESCRIPTIO
1. Information on the academic program
1.1.Higher education institution
1.2.Faculty
1.3.Department
1.4.Field
1.5.Study cycle
1.6.Program / Qualification
Spiru Haret University
Faculty of Mathematics and Informatics
Department of Mathematics and Informatics
Mathematics
Undergraduate
Mathematics
2. Information concerning the course subject
2.1.Name of subject
2.2 Discipline code
2.3.Course organizer
2.4.Seminar organizer
2.5. Year of study
Arithmetic and Number Theory
MI/MAT/3/4
Prof. univ. dr. Ion D. Ion
Lect. dr. Vlad Copil
2.6.Semester 1 2.7.Evaluation type
3
ES
2.8.Course type
O
3. Estimated time (hours per semester) of teaching / learning activities
3.1 No. hours per week
4
3.4 o. hours in the curriculum
56
3.2 of which course
hours:
3.5 of which course
hours:
2
28
3.3 of which seminar
/ lab hours:
3.6 of which seminar
/ lab hours:
Distribution of teaching / learning time
2
28
hours
Study of textbook, syllabus, bibliography and course notes
Further study in library, on electronic platforms, fieldwork
Preparation of seminars / labs, home assignments, papers, portfolio, essays
Tutoring
Examinations
Other………
3.7 Total hours of individual study
94
3.9 Total hours per semester
150
3.10 o. of credits
6
30
30
28
2
4
4. Prerequisites (where relevant)
4.1 curriculum-related
4.2 competence-related
•
•
•
Algebra 1
Working with concepts and mathematical methods
Demonstrate results using different mathematical
mathematical reasoning
concepts
5. Facilities and equipment (where relevant)
5.1. for the course
5.2. for the seminar / lab
•
•
Classroom equipped as required
Seminar room equipped as required
Professional
competences
6. Competences acquired during / after the course
•
•
•
•
Working with concepts and mathematical methods
Develop and analyze algorithms for solving
Demonstrate results using different mathematical concepts and mathematical reasoning
Applying mathematical models to solve interdisciplinary problems
and
Transversal
competences
•
not applicable
7. Course objectives (as resulting from the matrix of specific competences)
7.1 Course goals
•
7.2 Course objectives
•
The course covers fundamental concepts and results of
Number Theory, and their applications in Cryptography and
Computer Algebra.
Students completing the course are expected to become
familiar with, and learn how to apply the fundamental
theorem of arithmetic, and the Euclidian algorithm, how to
compute the greatest common divisor and Bezour
coefficients. Students will be introduced to, and learn how to
use the Mobius Inversion Formula, Fermat’s Little Theorem
and the Chinese Remainder Theorem. Students will also learn
how to use modular arithmetic in Cryptography.
8. Contents
8.1 Course
Teaching methods
Observations
Natural numbers. The principle of induction, the well- Lecture
ordering principle
Divisibility, the greatest common divisor, properties
Lecture
Prime numbers. The fundamental theorem of Lecture
arithmetic
The (extended) Euclidian Algorithm. Applications
Lecture
Euler's function, Mobius function and Mobius Lecture
inversion formula
Bertrand’s conjecture, Chebyshev’s theorem and some Lecture
comments on the Prime Number Theorem
Congruences, basic properties of congruences
Lecture
Linear congruences, Chinese remainder theorem
Lecture
Euler’s Theorem, Fermat's Little Theorem
Lecture
Polynomial congruence modulo a prime number Lecture
(Legendre's theorem) and modulo a prime power
Quadratic residues, Legendre's symbol. Properties
Lecture
The quadratic reciprocity law
Lecture
Finite (infinite) continued fractions. Properties
Lecture
Representation of an irrational number as infinite Lecture
simple continued fraction
Bibliography
1. Ion D Ion, S. Barza - Aritmetica, teoria numerelor si metode algoritmice în algebra, Editura FRM (2008)
2. Ion D Ion , C. Nită - Elemente de Aritmetică, Ed. Tehnică, 1978.
3. V. Alexandru, N. Gosoniu - Elemente de teoria numerelor, Tipografia Universitătii Bucuresti, 1999.
4. Allenby and Redfen – Introduction to Number Theory with Computing, London, 1989
5. M.Mignotte – Mathematics for computer algebra, Springer, 1991
6. V. Shoup – A Computational Introduction to Number Theory and Algebra, Cambridge Univ. Press, 2005
7. N. Koblitz, A course in Number Theory and Cryptography, Ed.Springer, 1998
8.2 Seminar / lab
Teaching methods
Observations
Computing the greates common divisor and the Exercises
Bezout coefficients
Computing the modular inverse of an integer
Exercises
Solving linear congruences
Exercises
Solving a system of linear congruences
Exercises
Modular representation of a positive integer
Exercises
Reducing the problem of solving of a polynomial Exercises
congruence to the case of a prime power modulus
Applications of the Mobius inversion formula
Exercises
Miller-Rabin primality test
Exercises
The fast powering algorithm modulo m
Exercises
The Jacobi symbol
Exercises
Primitive roots modulo m and discrete logarithms
Exercises
Diffie-Hellman protocol and public key criptosystems Exercises
Continued fractions and rational approximation of an Exercises
irrational number
Continued fractions and Pell’s equation
Exercises
Bibliography
1. Ion D Ion, S. Barza - Aritmetica, teoria numerelor si metode algoritmice în algebra, Editura FRM (2008)
2. Ion D Ion , C. Nită - Elemente de Aritmetică, Ed. Tehnică, 1978.
3. V. Alexandru, N. Gosoniu - Elemente de teoria numerelor, Tipografia Universitătii Bucuresti, 1999.
4. Allenby and Redfen – Introduction to Number Theory with Computing, London, 1989
5. M.Mignotte – Mathematics for computer algebra, Springer, 1991
6. V. Shoup – A Computational Introduction to Number Theory and Algebra, Cambridge Univ. Press, 2005
7. N. Koblitz, A course in Number Theory and Cryptography, Ed.Springer, 1998
9. Course’s relevance to the epistemic community, professional associations, and representative
employers in fields significant for the program
•
Number Theory is an important tool that high school teachers can use in shaping the mathematical thinking of
students. Number Theory also has applications in computer Science.
10. Assessment
Activity
10.1 Assessment criteria
10.3 Weight in the
final grade
10.4 Course
Involvement in lecture Record the frequency and strength
20%
with
questions, of interaction in the classroom..
comments, examples of
analysis.
10.5 Seminar / lab
Involvement
in
the Record the frequency and strength
20%
preparation
and of interaction between the hours
discussion of problems
of seminar.
10.6 Minimal performance standard
• Students will be required to operate correctly with prime numbers and the fundamental theorem of
arithmetic. They will be expected to develop a solid understanding of principal algorithmic methods of
arithmetic and will be able to use correctly Fermat’s Little Theorem.
Date:
………………….
Date of Dept. approval
………………………
10.2 Assessment method
Course organizer’s signature,
Seminar organizer’s signature,
………………………………
…………………………………..
Head of Dept. signature
……………………………………