THE COURSE WILL BE ONLINE THIS TERM until announced otherwise by the university.
FOLLOW THE ONLINE LECTURES AND MATERIALS IN THE ONLINE EDUCATION SYSTEM OF DEU USING SAKAI, you will find there lecture recordings and some supporting videos for our course:
https://online.deu.edu.tr
The documents, homeworks etc. for the course that we have given/assigned are the below files ordered by date.
ELEMENTARY NUMBER THEORY
For elementary number theory, we shall follow the book
Silverman, J. H. A Friendly Introduction to Number Theory. Pearson, 2012.
See The Babylonian tablet Plimpton 322, dating 1800 BCE, which contains large Pyhagorean triples: "It seems plausible, therefore, that the Babylonians knew how to generate primitive Pythagorean triples." See also the wikipedia article on Plimpton 322.
Worksheet for Pythagorean Triples and Rational Points on Curves
A very useful web page for the history of mathematics:
MacTutor History of Mathematics archive.
See there Pierre de Fermat. What is Fermat's Last Theorem?
Divisibility and the Sieve of Eratosthenes
Sieve of Eratosthenes to find primes less than 100
Worksheet for the method of Sieve of Eratosthenes
Division algorithm and Euclidean algorithm to find gcd(a,b)=as+bt for some integers s,t
Worksheet for Euclidean Algorithm, Linear Equations and Linear Congruences
Proof of Lame's Theorem for the number of steps in Euclidean algortihm: The number of divisions in the Euclidean algorithm for two positive integers is at most 5 times the number of decimal digits of the smaller positive integer.
Prime divisibility, Fundamental Theorem of Arithmetic, gcd(a,b) and lcm(a,b)
gcd, Fundamental Theorem of Arithmetic and Modular Arithmetic
See the following notes for the algebraic view of Fermat's Little Theorem, Euler-Fermat Formula and the Chinese Remainder Theorem:
The ring Zn of integers modulo n
Rings, integral domains and fields
When is the ring Zn an integral domain? A field?
Euler's phi function and the Chinese Remainder Theorem
The group Un of units in Zn and orders of its elements
Groups and orders of its elements
Using Lagrange's Theorem for groups to prove Euler-Fermat formula and Fermat's Little Theorem
Elementary Number Theory Exercises
Midterm and Final Examinations of 2017 and 2019
An example for taking powers modulo m by successive squaring
Worksheet for Computing k-th roots modulo m
Subjects of the Midterm Examination
Till now we have studied the first 18 chapters (pages 1-136) of your first textbook (Silverman, J. H. A Friendly Introduction to Number Theory. Pearson, 2014) for elementary number theory. In the next weeks, we shall study some topics from your other textbooks.
COMPLEX NUMBERS
Complex Numbers Worksheet. Solve the problems in this worksheet and study pages 160--182 for complex numbers from your second textbook:
Rotman, J. J. A Journey into Mathematics, An Introduction to Proofs. Dover, 2007.
SEQUENCES OF REAL NUMBERS
Study Sections 1.1, 1.2, 1.3, Appendix A for a review of the Axiomatic Construction of the Real Number System and Completeness Property of R, and Sections 2.1, 2.2, 2.3, 2.4 for sequences of real numbers from the following analysis book:
Wade, W. R. Introduction to Analysis. 4th edition, new international edition. Pearson, 2014.
See also the web page of the course MAT1015 Technical English I for a review of the Axiomatic Construction of the Real Number System.
Sum, Product and Quotient of Sequences, and Further Important Theorems for Sequences (try to prove these theorems in these notes using the definition of convergence of sequences) (and in presentation form)
Exercises for Sequences of Real Numbers
True/False Questions for Sequences (Be sure that you can do very well these true/false questions for sequences. It will be a good review for sequences.)
Worksheet for Sequences of Real Numbers
Subjects of the Final Examination with suggestions for summer study to gain some mathematical maturity that you shall need more in your second year in our department.