Course Overview
This course provides an introduction to elementary number theory, one of the oldest and most beautiful branches of mathematics. We will explore the concepts of divisibility, prime numbers, congruences, and their applications to modern cryptography.
Topics covered include:
- Pythagorean triples and Diophantine equations
- Prime numbers and the Fundamental Theorem of Arithmetic
- Greatest common divisors and the Euclidean algorithm
- Mathematical proof techniques: direct proof, contradiction, and induction
- Modular arithmetic and congruences
- Fermat's Little Theorem and Euler's Theorem
- The Chinese Remainder Theorem
- Applications to cryptography: RSA and Diffie-Hellman algorithms
- Quadratic residues and quadratic reciprocity
- Primitive roots
- Computational methods using SageMath
Learning Outcomes
By the end of this course, students will be able to:
- Develop problem-solving skills and mathematical reasoning abilities
- Understand foundational concepts in number theory including divisibility, primes, factorization, and congruences
- Construct rigorous mathematical proofs using direct proof, proof by contradiction, and mathematical induction
- Apply the Euclidean algorithm to find greatest common divisors and solve linear Diophantine equations
- Work with modular arithmetic and solve congruence equations
- Understand and apply important theorems in modular arithmetic including Fermat's Little Theorem, Euler's Theorem, the Chinese Remainder Theorem, and existence of primitive roots
- Comprehend the mathematical foundations of modern cryptographic systems including RSA and Diffie-Hellman
- Use computational tools (SageMath) to explore number-theoretic concepts and verify results
Grade Breakdown
| Midterm 1 | 20% | (Week 5: Feb 18-20) |
| Midterm 2 | 20% | (Week 10: Mar 31-Apr 2) |
| Final Exam | 28% | |
| Problem Sets | 30% | (Due Fridays at 5:00 PM) |
| Section Attendance | 2% | |
| Total | 100% |
Problem Sets
Problem sets will be assigned weekly and are due on Fridays at 5:00 PM. Problem sets are an essential component of the course and provide practice with the concepts and techniques covered in lecture.
Late Policy
Late work may be accepted on a case-by-case basis, but there must be a good reason for the late submission. Please communicate with the instructor as soon as possible if you anticipate difficulty meeting a deadline.
Important: After one week, homework will not be accepted for any reason. Plan accordingly and don't wait until the last minute to start your assignments.
Students are encouraged to work together on problem sets and discuss approaches to problems. However, each student must write up their own solutions independently. Copying solutions from other students or online sources is not permitted.
Midterms
Midterms will be held in class. Students are allowed to use calculators during the exam. However, the exams are closed book and closed notes. The exams will be cumulative, covering all material discussed up to the exam date.
Textbook
Required: Joseph H. Silverman, A Friendly Introduction to Number Theory (4th Edition)
Reading assignments from the textbook will be posted on the course schedule. Additional resources and materials will be provided through the course website.
Attendance and Participation
Lecture: Tuesday/Thursday, 9:00-10:15 AM in Bloomberg 168
Discussion Section: Friday, 9:00-9:50 AM in Bloomberg 276
Regular attendance at both lectures and discussion sections is expected. Discussion sections provide an opportunity to work through problems, ask questions, and solidify your understanding of the material. Section attendance counts for 2% of your final grade.
Note: Week 7 (March 3-5) will be conducted remotely. Details will be provided closer to the date.
Additional Resources
We will use SageMath throughout the course for computational exploration of number-theoretic concepts. No prior programming experience is required; tutorials and examples will be provided on the course website.
Office hours for the instructor and TA are listed on the course homepage. Please take advantage of office hours if you have questions or need additional help with the material.
Academic Integrity
Students are expected to adhere to the Johns Hopkins University Academic Integrity policies (please see the syllabus on Canvas for a detailed statement). All work submitted must be your own. While collaboration on problem sets is encouraged, you must write up your solutions independently. Any violations of academic integrity will be taken seriously and reported to the appropriate university authorities.
AI Policy
The use of AI tools, such as ChatGPT or similar, is permitted for brainstorming and general guidance. However, all submitted work must be your own. Direct copying of AI-generated content is not allowed and will be considered a violation of academic integrity.
Note: I strongly discourage reliance on AI tools. They are frequently incorrect, and if you rely on them without attempting the problems yourself, you will likely struggle on the midterms and not perform well.