| MATHEMATICS | |||||
| Bachelor | TR-NQF-HE: Level 6 | QF-EHEA: First Cycle | EQF-LLL: Level 6 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT3010 | Number Theory | Spring | 3 | 0 | 3 | 5 |
| The course opens with the approval of the Department at the beginning of each semester |
| Language of instruction: | En |
| Type of course: | Must Course |
| Course Level: | Bachelor |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Instructor NERMINE AHMED EL SISSI |
| 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. |
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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. |
| 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. |
| 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 | ||
| Course Notes: | 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. |
| Semester Requirements | Number of Activities | Level of Contribution |
| Attendance | 16 | % 0 |
| Laboratory | % 0 | |
| Application | % 0 | |
| Field Work | % 0 | |
| Special Course Internship (Work Placement) | % 0 | |
| Quizzes | 2 | % 20 |
| Homework Assignments | % 0 | |
| Presentation | % 0 | |
| Project | % 0 | |
| Seminar | % 0 | |
| Midterms | 2 | % 40 |
| Preliminary Jury | % 0 | |
| Final | 1 | % 40 |
| Paper Submission | % 0 | |
| Jury | % 0 | |
| Bütünleme | % 0 | |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 60 | |
| PERCENTAGE OF FINAL WORK | % 40 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Laboratory | 0 | 0 | 0 |
| Application | 0 | 0 | 0 |
| Special Course Internship (Work Placement) | 0 | 0 | 0 |
| Field Work | 0 | 0 | 0 |
| Study Hours Out of Class | 14 | 2 | 28 |
| Presentations / Seminar | 0 | 0 | 0 |
| Project | 0 | 0 | 0 |
| Homework Assignments | 0 | 0 | 0 |
| Quizzes | 7 | 2 | 14 |
| Preliminary Jury | 0 | 0 | 0 |
| Midterms | 2 | 10 | 20 |
| Paper Submission | 0 | 0 | 0 |
| Jury | 0 | 0 | 0 |
| Final | 1 | 20 | 20 |
| Total Workload | 124 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution |