Course Name Code Semester T+U Hours Credit ECTS
Diophantine Equations MAT 585 0 3 + 0 3 6
 Precondition Courses Recommended Optional Courses Course Language Turkish Course Level yuksek_lisans Course Type Optional Course Coordinator Dr.Öğr.Üyesi BAHAR DEMİRTÜRK BİTİM Course Lecturers Course Assistants Research asistants of mathematics department Course Category Course Objective Introducing Diophantine equations and finding the answer of the question" Is given Diophantine equation be solvable? If it is solvable, are its solutions finite or infinite? What are the all solutions of this equation?". All solutions of Diophantine equations will be obtained by different methods, with uzing number theory and algebra knowledge. Moreover our aim is investigating the relation between Fibonacci-Lucas numbers and Diophantine equations, Pell-Pell Lucas numbers and Diophantine equations. Course Content Definition of Diophantine equations, elementary Methods for Solving Diophantine equations; The Factoring Method, using Inequalities, the parametric method, the modular arithmetic method, the method of mathematical induction, Fermats method of infinite descent, some classical Diophantine equations; linear Diophantine equations, Pythagorean triples and related problems, Pell-Type equations, solving Pells equation, the equation , some advanced methods for solving Diophantine equations; the ring of Gaussian integers, the ring of integers of , Quadratic Reciprocity and Diophantine equations, divisors of certain forms such as a^2+b^2, a^2+2b^2 and a^2-2b^2.
# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/she recognizes Diophantine equations. Lecture, Question-Answer, Self Study, Problem Solving, Testing, Oral Exam, Homework,
2 He/she learns solving methods of Diophantine equations. Lecture, Question-Answer, Self Study, Problem Solving, Testing, Oral Exam, Homework,
3 He/she recognizes Pell equations and investigates solutions of them. Lecture, Question-Answer, Self Study, Problem Solving, Testing, Oral Exam, Homework,
4 He/she learns properties of Gauss integers ring Z[i]. Lecture, Question-Answer, Self Study, Problem Solving, Testing, Oral Exam, Homework,
5 He/she learns properties of integers ring Q(d^1/2). Lecture, Question-Answer, Self Study, Problem Solving, Testing, Oral Exam, Homework,
6 He/she understands the relation between Legendre symbol and solutions of Diophantine equations. Lecture, Question-Answer, Self Study, Problem Solving, Testing, Oral Exam, Homework,
Week Course Topics Preliminary Preparation
1 Definition of Diophantine equations.
2 Some elementary methods for solving Diophanitne equations; decomposition method, inequalities, parametric method.
3 Modular arithmetic and induction method.
4 Fermats infinite descent method.
5 Some classical Diophantine equations: Linear Diophantine equations.
6 Pisagor triples and related problems.
7 Pell equations.
8 Solutions of Pell equations.
9 Solutions of the equation ax^2-by^2=1.
10 Some further methods for solutions of Diophantine equations.
11 Gauss integers ring Z[i].
12 Integers ring Q(d^1/2).
13 Legendre symbol and Diophantine equations.
14 Divisors of the form a^2+b^2, a^2+2b^2 and a^2-2b^2.
Resources
Course Notes
Course Resources Fibonacci and Lucas Numbers and the Golden Section, S. Vajda, Ellis Horwood Limt. Publ., England, 1989.

An Introduction to Theory of Numbers, Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery, Wiley, 1991.

Number Theory Volume I: Tools and Diophantine Equations, Henri Cohen, Springer, 2007.

An Introduction to Diophantine Equations: A Problem Based Approach Book, Titu Andrescu, Dorin Andrica, Ion Cucurezeanu, Birkhouse, 2010.
Order Program Outcomes Level of Contribution
1 2 3 4 5
0 Develop strategic, political and practice plans and evaluate the results by taking into account the quality process in his/her area of expertise X
2 Student follows the current journals in his/her field and puts forward problems. X
3 Student understands the relations between the disciplines pertaining to the undergraduate programs of Mathematics X
4 Student gets new knowledge by relating the already acquired experience and knowledge with the subject-matters out of his/her field. X
5 Student uses different proof methods to come to a solution by analyzing the problems encountered. X
6 Student determines the problems to be solved within his/her field and if necessary takes the lead. X
7 Student conveys, in team work, his/her knowledge in the studies done in different disciplines by applying the dynamics pertaining to his/her own field. X
8 Student critically evaluates the knowledge got at the bachelor´s degree level, makes up the missing knowledge and focuses on the current subject-matters X
9 Student knows a foreign language to communicate orally and in writing and uses the foreign language in a way that he/she can have a command of the Maths terminology and can do a source research. X
10 Student improves himself/herself at a level of expertness in Mathematics or in the fields of application by improving the knowledge got at the bachelor´s degree level. X
11 Student considers the scientific and cultural ethical values in the phases of gathering and conveying data or writing articles. X
Evaluation System
Semester Studies Contribution Rate
1. Ara Sınav 70
1. Ödev 30
Total 100
1. Yıl İçinin Başarıya 50
1. Final 50
Total 100