COURSE UNIT TITLE

: NUMBER THEORY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4064 NUMBER THEORY ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASISTANT PROFESSOR SALAHATTIN ÖZDEMIR

Offered to

Mathematics (Evening)
Mathematics

Course Objective

The aim of this course is to introduce elementary number theory and form a bridge to geometric and algebraic methods in number theory by studying quadratic reciprocity, quadratic forms, Pell s equation and its solution by continued fraction expansions, units in quadratic number fields, the ring of Gaussian integers, and rational points on elliptic curves.

Learning Outcomes of the Course Unit

1   Will be able to solve linear congruence systems using greatest common divisors and Euclidean algorithm.
2   Will be able to find integer solutions of some Diophantine equations.
3   Will be able to determine quadratic residues using the Quadratic Reciprocity Law.
4   Will be able to find solutions of Pell s equation using periodic continued fraction expansions of square roots of positive integers.
5   Will be able to interpret the relation between the geometric idea of finding rational points on curves and solutions of Diophantine equations.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Reviewofelementarynumbertheory: Diophantineequations. Pythagoreantriplesandtheunitcircle. Fermat slasttheorem. Divisibilityandthegreatestcommondivisor. Euclideanalgorithm. Linearequations. Factorizationsintoprimes and the FundamentalTheoremofArithmetic. Congruences. Fermat s LittleTheorem.Euler s formulaandEuler sPhifunction.
2 Powers modulo m and successive squaring (modular exponentiation). Computing kth roots modulo m. Powers, roots and unbreakable codes, RSA public key cryptosystem. Primality testing and Carmichael numbers. Sums of divisors.
3 Powers modulo a prime p and primitive roots, indices. Primitive roots and power residues.
4 Quadratic residues and the law of quadratic reciprocity. The Jacobi symbol.
5 Numbers that are sums of two squares.Quadratic forms. The four square theorem.
6 Pell s equation. Diophantine approximation. Continued fractions.
7 Periodic continued fractions and Pell s equation.
8 Irrational numbers and transcendental numbers. Best possible approximations. Farey fractions and irrational numbers.
9 Midterm
10 The ring of Gausssian integers and the unique factorization into irreducibles in this ring.
11 Multiplicative number theoretic functions, the Möbius inversion formula.
12 Rational points on curves, cubic curves. A rough sketch of the ideas in the proof of Fermat s last theorem.
13 Elliptic Curves with Few Rational Points. Points on Elliptic Curves Modulo p. Torsion Collections Modulo p and Bad Primes. Defect Bounds and Modularity Patterns. Elliptic Curves and Fermat s Last Theorem.
14 Motivation for Algebraic Number Theory: Pell s equation and units in quadratic number fields, failure of unique factorization into irreducibles in some ring of algebraic integers.

Recomended or Required Reading

Textbook(s):
1) Silverman, J.H. A Friendly Introduction to Number Theory, Third edition, Pearson, 2006.
2) Niven, I., Zuckerman, H. S., and Montgomery, H. L.An Introduction to The Theory of Numbers, Fifth edition, John Wiley & Sons, 1991.
Supplementary Book(s):
1) Stillwell, J. Elements of Number Theory, Springer, 2003.
2) Long, C. T.Elementary Introduction to Number Theory, Thirrd edition, Prentice Hall, Inc., Englewood Cliffs, 1987.
3) Childs, L. N. A Concrete Introduction to Higher Algebra, Second edition, Springer, 1995.
References:
1) Scharlau, W. and Opolka, H. From Fermat to Minkowski, Lectures on the Theory of Numbers and Its Historical Development, Springer-Verlag, 1985.
2) Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, Second edition, Springer, 1990.
3) Nathanson, M. B. Elementary Methods in Number Theory, Springer, 2000.
4) Silverman, J.H. and Tate, J.Rational Points on Elliptic Curves, Springer, 1992.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving, Discussion

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 FIN FINAL EXAM
3 FCG FINAL COURSE GRADE MTE * 0.40 + FIN * 0.60
4 RST RESIT
5 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.40 + FIN * 0.60


Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr
Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 13 4 52
Preparations before/after weekly lectures 12 4 48
Preparation for midterm exam 1 30 30
Preparation for final exam 1 30 30
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 166

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.1535333
LO.2535333
LO.3535333
LO.4535333
LO.5535333

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

Copyright 2015 DEÜ. BİD

HOME / ABOUT US / DEGREE PROGRAMMES / GENERAL INFORMATION FOR STUDENTS / INTERNATIONAL RELATIONS