Elementary Number Theory (初等整数論)
(Nagoya University, Special Mathematics Lecture, Fall 2025)
This course is an introduction to number theory and is intended for undergraduate students across disciplines (G30 Program, Physics, Engineering, Computer Science, Mathematics). NUPACE students are very welcome. Japanese students who wish to take a course taught in English can receive support through the NU-EMI Project.
本講義は数論の入門であり、学部生（G30プログラム、物理、工学、コンピュータサイエンス、数学など）を対象としています。NUPACEの学生も歓迎します。英語で開講される授業の受講を希望する日本人学生には、NU-EMIプロジェクトによるサポートがあります。初回講義は10月6日に実施します。
Materials
Place & Time
Grading & Homework submission:
Based on homework assignments and a final exam.
The credits follow the new rule for SML classes. (SML=Special mathematics lecture)
Please contact me via email if you have any questions.
References
Books:
The following gives an overview of the topics we will cover each week.
Week 02 (10/06-10/18): Introduction: What is Number Theory?
Week 03 (10/13-10/19): Problems in Number Theory: Twin primes, Sum of squares
Week 04 (10/20-10/26): Divisibility & Primes
Week 05 (10/27-11/02): Arithmetic functions I
Week 06 (11/03-11/09): No lecture on Monday, 3rd November (Culture day)
Week 07 (11/10-11/16): Arithmetic functions II & L-Functions, Congruences I
Week 08 (11/17-11/23): Congruences II
Week 09 (11/24-11/30): Chinese Remainder Theorem
Week 10 (12/01-12/07): Chinese Remainder Theorem II & Units in Z/mZ
Week 11 (12/08-12/14): Primitive roots and cyclic groups mod p
Week 12 (12/15-12/21): Primitive roots and cyclic groups mod p II
Week 13 (12/22-12/29): 🎅 Number Theory Christmath Challenge 🎅
☃️ Winter Vacation (12/28-01/07) ☃️
Week 14 (01/05-01/11): TBA
Week 15 (01/12-01/18): TBA
Week 16 (01/19-01/25): TBA
Week 17 (01/26-02/06): Final Exam (26th January, during the lecture)
Last update: 9th December 2025.
本講義は数論の入門であり、学部生（G30プログラム、物理、工学、コンピュータサイエンス、数学など）を対象としています。NUPACEの学生も歓迎します。英語で開講される授業の受講を希望する日本人学生には、NU-EMIプロジェクトによるサポートがあります。初回講義は10月6日に実施します。
Materials
- Course overview
- Overview notes (v. 4, 1st December 2025)
- Handwritten lecture notes: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9
- Homework: Homework 1, Homework 2, Homework 3
- Overview of multiplicative arithmetic functions
- Overview of classical and modern Number Theory
- Primes and factorization: fundamental theorem of arithmetic
- Fermat–Euler–Wilson: Euler’s totient, little theorems, and applications
- Congruences: modular arithmetic, Chinese Remainder Theorem
- Primitive roots and cyclic groups mod p
- Quadratic residues: Legendre symbol, Gauss lemma, Quadratic Reciprocity
- Basics of Algebraic Number Theory
- Basics of Modular forms & Elliptic Curves
- Zeta & L-functions
Place & Time
- Lecture: Monday 6th period (18:15-19:45). First lecture: October 6th
- Place: C15 (Liberal Arts and Sciences Main Building)
Grading & Homework submission:
Based on homework assignments and a final exam.
The credits follow the new rule for SML classes. (SML=Special mathematics lecture)
Please contact me via email if you have any questions.
References
Books:
- [IR] K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory, 2nd ed., Graduate Texts in Mathematics 84, Springer, New York (1990).
The following gives an overview of the topics we will cover each week.
Week 02 (10/06-10/18): Introduction: What is Number Theory?
Week 03 (10/13-10/19): Problems in Number Theory: Twin primes, Sum of squares
Week 04 (10/20-10/26): Divisibility & Primes
Week 05 (10/27-11/02): Arithmetic functions I
Week 06 (11/03-11/09): No lecture on Monday, 3rd November (Culture day)
Week 07 (11/10-11/16): Arithmetic functions II & L-Functions, Congruences I
Week 08 (11/17-11/23): Congruences II
Week 09 (11/24-11/30): Chinese Remainder Theorem
Week 10 (12/01-12/07): Chinese Remainder Theorem II & Units in Z/mZ
Week 11 (12/08-12/14): Primitive roots and cyclic groups mod p
Week 12 (12/15-12/21): Primitive roots and cyclic groups mod p II
Week 13 (12/22-12/29): 🎅 Number Theory Christmath Challenge 🎅
☃️ Winter Vacation (12/28-01/07) ☃️
Week 14 (01/05-01/11): TBA
Week 15 (01/12-01/18): TBA
Week 16 (01/19-01/25): TBA
Week 17 (01/26-02/06): Final Exam (26th January, during the lecture)
Last update: 9th December 2025.