IntroductIon to Number Theory
IntroductIon to Number Theory
Name: MAT 364 - Introduction to Number Theory.
Time and Place: 13:00 / 16:00, H307.
Midterm Date: May 24, 2025.
Final Date: TBA.
Instructor Office Hours: Mondays 11.00-13.00, H302B (by appointment).
Main Reference: An Introduction to the Theory of Numbers-Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery. S, 5th Edition (NZM)
Click here for Syllabus.
Möbius Inversion and Dirichlet Convolution Lecture Notes
Multiplicative Functions Review Notes
Midterm (due Friday, May 2nd, 5 pm).
Midterm Solutions (Posted 06/05/2025).
You can turn in your homework in class or via e-mail. Don't forget to use the answer sheet to check your answers and add detailed corrections before turning it in. Assignments will not be graded on correctness.
HW 1 (Due 17.02.2025): NZM 1.2 # 3a, 3d, 3e, 15,16,17,18, 25, 33, 50, 53.
+ Generalize the statement (a,b) is the minimum positive value ax+by to (a1,a2,...,ak) and prove it.
Answer Sheet: Here. (Posted 13/02/25, corrected 28/02/25)
HW 2 (Due 24.02.2025): NZM 1.3 # 22, 26 (part 2), 27, 48. NZM 1.4 #21, 22.
+ Exercises on Binomial Coefficients.
Answer Sheet: Here. (Posted 21/02/25)
HW 3 (Due 03.03.2025): NZM 2.1 # 8, 9, 19, 20, 23, 27, 32, 33, 40. NZM 2.2 # 8,9.
Answer Sheet: Here. (Posted 28/02/25)
HW 4 (Due 10.03.2025): NZM 2.1 #48. NZM 2.2 # 5 (parts a, b, d, g only), 6. NZM 2.3 # 3, 4, 7, 15, 45. NZM 2.7 #6.
Answer Sheet: Here. (Posted 06/03/25)
HW 5 (Due 17.03.2025): NZM 2.3 # 29, 30. NZM 2.6 #2, 3, 10. NZM 2.8 # 1, 3, 21, 24.
Answer Sheet: Here. (Posted 14/03/25)
HW 6 (Due 28.03.2025): NZM 2.8 # 8, 12. NZM 4.1 # 1, 3, 6.
Answer Sheet: Here. (Posted 17/03/25)
HW 7 (Due 21.04.2025): NZM 4.2 # 5, 6, 10, 20, 22, 24.
+ Calculate the sum Σ_{d|n} f(d) for arithmetic functions d(n) and σ(n).
Answer Sheet: Here. (Posted 18/04/25)
HW 8 (Due 05.05.2025): NZM 4.3 # 1, 2, 3, 7, 8, 15, 17, 22.
+ Show that the convolution operation is associative.
Answer Sheet: Here. (Posted 06/05/25)
HW 9 (Due 12.05.2025): NZM 5.1 Find all integer solutions and all positive integer solutions for equations from # 2, 3c, 4a, 4b. NZM 5.2 # 1, 2. NZM 5.3 # 1, 2, 3, 10.
Answer Sheet: Here. (Posted 12/05/25)
HW 10 (Due 23.05.2025): NZM 7.1 # 1, 3, 4. NZM 7.3 # 2, 3b, 3c.
+ Prove that no rational number can be approximated to order 2.
Hint: Assume that for a rational number x=a/b,
|x-p/q|<C/(q^2).
Try to find an upper limit for q in terms of C and b.
Answer Sheet: Here. (Posted 13/05/25).
(Updated on 05.05.2025)
Part 1- Divisibility: (3 weeks).
The Euclidean Algorithm, greatest common divisor, prime numbers, the Fundamental Theorem of Algebra, Binomial Expansion.
(NZM: 1.2, 1.3, 1.4)
Part 2 - Modular Arithmetics: (3/4 weeks).
Modular Structures. Fermat’s theorem. Solutions of modular equations. Chinese Remainder Theorem. Prime Moduli. Prime Power Moduli. Primitive Roots.
(NZM: 2.1, 2.2, 2.3, 2,6, 2.7, 2.8)
Extra Topic: Cryptography (NZM: 2.4)
Part 3- Number Theoretical Functions: (2/3 weeks).
Greatest Integer Function. Multiplicative Functions. Möbius Inversion. (NZM: 4.1, 4.2, 4.3)
Extra Topic: Dirichlet Convolution.
Extra Topic: Combinatorial Methods in Number Theory (NZM: 4.5)
Part 4- Diophantine Equations: (2 weeks).
Linear Equations. Markov Equation and Vieta Jumps. (NZM: 5.1, 5.2, 5.3)
Extra Topic: Elliptic Curves (NZM: 5.7)
Part 5- Approximations with Rationals: (2/3 weeks).
Farey Tree. Approximating Real Numbers with Rationals. Continued Fractions. (NZM: 6.1, 6.2, 7.1, 7.2)
Extra Topic: Irrationality (NZM: 6.3, 7.4)