MATHEMATICS
Bachelor	TR-NQF-HE: Level 6	QF-EHEA: First Cycle	EQF-LLL: Level 6

Course Introduction and Application Information

Course Code	Course Name	Semester	Theoretical	Practical	Credit	ECTS
MAT3010	Number Theory	Spring	3	0	3	5

Basic information

Language of instruction:	English
Type of course:	Must Course
Course Level:	Bachelor’s Degree (First Cycle)
Mode of Delivery:	Face to face
Course Coordinator :	Instructor NERMINE AHMED EL SISSI
Recommended Optional Program Components:	None
Course Objectives:	The aim of this course is to introduce students to some basic ideas of number theory. The course will introduce different methods of proof that students can apply within the context of elementary number theory. This will enable students to witness the development of mathematics through creating examples, building conjectures, validating these conjectures via proofs to obtain theorems.

Learning Outcomes

The students who have succeeded in this course;
• Use different methods of proof to verify mathematical statements, such as proof by induction, by contraposition and by contradiction.
• understand the basics of modular arithmetic.
• Introduce Euler Totient function as an example of multiplicative functions.
• State and prove Fermat's Little Theorem, Euler’s Theorem and explore some of their applications.
• solve systems of Diophantine equations using the Euclidean algorithm and the Chinese Remainder Theorem.
• Study quadratic polynomial congruences and apply Legendre symbols to examine the existence of solution.
• Define primitive roots and understand their role in simplifying modular arithmetic.
• Define Pythagorean triples and show how to generate them.

Course Content

The course covers the following topics: divisibility, the Fundamental Theorem of Algebra, congruences, arithmetic functions, Euler Totient function, polynomial congruences, quadratic residues and the Legendre symbol, the Jacobi symbol, primitive roots, and Pythagorean Triples.

Weekly Detailed Course Contents

Week	Subject	Related Preparation
1)	Divisibility, the Fundamental Theorem of Arithemtic, Euclidean Algorithm
2)	Modular Arithmetic and their properties
3)	Modular arithmetic continued, polynomial congruences
4)	polynomial congruences and the Chinese Remainder Theorem
5)	Mathematical induction revisited, arithmetic functions
6)	Multiplicative arithmetic functions, and Fermat's Little Theorem
7)	Wilson's Theorem and quadratic residues
8)	Quadratic residues
9)	Legendre symbol and Euler's criterion
10)	Gauss Quadratic Reciprocity Law
11)	Pseudoprimes
12)	Primitive roots
13)	Primitive roots continued
14)	Pythagorean Triples

Sources

Course Notes / Textbooks:	A Friendly Introduction to Number Theory, Joseph H. Silverman, Pearson 4th Edition 2014.
References:	Elementary Number Theory and Its Applications, K.H. Rosen, (4th edition) Addison-Wesley 2000.

Evaluation System

Semester Requirements	Number of Activities	Level of Contribution
Attendance	16	% 0
Quizzes	2	% 20
Midterms	2	% 40
Final	1	% 40
Total		% 100
PERCENTAGE OF SEMESTER WORK		% 60
PERCENTAGE OF FINAL WORK		% 40
Total		% 100

ECTS / Workload Table

Activities	Number of Activities	Duration (Hours)	Workload
Course Hours	14	3	42
Study Hours Out of Class	14	2	28
Quizzes	7	2	14
Midterms	2	10	20
Final	1	20	20
Total Workload			124

Contribution of Learning Outcomes to Programme Outcomes

No Effect	1 Lowest	2 Low	3 Average	4 High	5 Highest

	Program Outcomes	Level of Contribution
1)	To have a grasp of basic mathematics, applied mathematics and theories and applications in Mathematics	4
2)	To be able to understand and assess mathematical proofs and construct appropriate proofs of their own and also define and analyze problems and to find solutions based on scientific methods,	4
3)	To be able to apply mathematics in real life with interdisciplinary approach and to discover their potentials,	4
4)	To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself,	4
5)	To be able to tell theoretical and technical information easily to both experts in detail and non-experts in basic and comprehensible way,	4
6)	To be familiar with computer programs used in the fields of mathematics and to be able to use at least one of them effectively at the European Computer Driving Licence Advanced Level,	4
7)	To be able to behave in accordance with social, scientific and ethical values in each step of the projects involved and to be able to introduce and apply projects in terms of civic engagement,
8)	To be able to evaluate all processes effectively and to have enough awareness about quality management by being conscious and having intellectual background in the universal sense,	4
9)	By having a way of abstract thinking, to be able to connect concrete events and to transfer solutions, to be able to design experiments, collect data, and analyze results by scientific methods and to interfere,
10)	To be able to continue lifelong learning by renewing the knowledge, the abilities and the competencies which have been developed during the program, and being conscious about lifelong learning,	4
11)	To be able to adapt and transfer the knowledge gained in the areas of mathematics ; such as algebra, analysis, number theory, mathematical logic, geometry and topology to the level of secondary school,	4
12)	To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively.	4