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response = « **Course on Number Theory**
**Module 1: Properties of Integers**
### Overview
In this module, we will explore the fundamental properties of integers and their relationships with each other. We will cover topics such as divisibility, prime numbers, and modular arithmetic.
### Topics Covered:
* Divisibility and factors
* Prime numbers and composite numbers
* Modular arithmetic and its applications
* Congruences and their properties
### Lecture Notes:
#### Day 1: Divisibility and Factors
* Introduction to divisibility and factors
* Definition of prime and composite numbers
* Examples and exercises
#### Day 2: Prime Numbers and Composite Numbers
* Properties of prime numbers (e.g. unique factorization)
* Applications of prime numbers (e.g. cryptography)
* Properties of composite numbers (e.g. even/odd, square/cube)
#### Day 3: Modular Arithmetic and Congruences
* Introduction to modular arithmetic
* Definition of congruence and its properties
* Examples and exercises
### Homework and Assignments:
1. Prove that every even integer greater than 2 is composite.
2. Write a program in your preferred programming language to find the first prime number greater than 1000.
3. Solve the following congruences: x ≡ 2 (mod 5), x ≡ 3 (mod 7)
### Assessment:
* Quizzes and exams will be held throughout the module to assess understanding of the material.
* A final project will be assigned, which will involve applying number theory concepts to a real-world problem or cryptography.
**Module 2: Theorems and Conjectures**
### Overview
In this module, we will explore some of the most famous theorems and conjectures in number theory. We will cover topics such as Fermat’s Last Theorem, the Prime Number Theorem, and the Riemann Hypothesis.
### Topics Covered:
* Fermat’s Last Theorem (FLT) and its history
* Proof of FLT using modular forms
* The Prime Number Theorem (PNT) and its applications
* The Riemann Hypothesis (RH) and its significance
### Lecture Notes:
#### Day 1: Fermat’s Last Theorem
* Introduction to FLT and its history
* Overview of the proof using modular forms
* Examples and exercises
#### Day 2: The Prime Number Theorem
* Introduction to the PNT and its applications
* Proof of the PNT using the method of contour integration
* Examples and exercises
#### Day 3: The Riemann Hypothesis
* Introduction to the RH and its significance
* Overview of the proof attempts (e.g. Hardy-Littlewood, Selberg)
* Examples and exercises
### Homework and Assignments:
1. Prove that FLT is equivalent to the statement « there are infinitely many prime numbers of the form 2^p + p ».
2. Write a program in your preferred programming language to generate a random prime number.
3. Solve the following problem: Show that the function ϕ(n) = n – σ(n), where σ(n) is the sum of divisors, is always even.
### Assessment:
* Quizzes and exams will be held throughout the module to assess understanding of the material.
* A final project will be assigned, which will involve applying number theory concepts to a real-world problem or cryptography.
**Module 3: Cryptography**
### Overview
In this module, we will explore the application of number theory in cryptography. We will cover topics such as RSA encryption, prime numbers in cryptographic applications, and cryptographic attacks.
### Topics Covered:
* Introduction to cryptography and its importance
* RSA encryption and its security
* Prime numbers in cryptographic applications (e.g. key exchange, digital signatures)
* Cryptographic attacks (e.g. factoring, discrete logarithm)
### Lecture Notes:
#### Day 1: Introduction to Cryptography
* Overview of cryptography and its importance
* History of cryptography and its evolution
#### Day 2: RSA Encryption
* Introduction to RSA encryption
* Proof of the security of RSA using number theory
* Examples and exercises
#### Day 3: Prime Numbers in Cryptographic Applications
* Introduction to prime numbers in cryptographic applications (e.g. key exchange, digital signatures)
* Proof of the security of these applications using number theory
* Examples and exercises
### Homework and Assignments:
1. Implement RSA encryption using a programming language.
2. Write a program in your preferred programming language to generate a random prime number for use in cryptography.
3. Solve the following problem: Show that the function ϕ(n) = n – σ(n), where σ(n) is the sum of divisors, is always even.
### Assessment:
* Quizzes and exams will be held throughout the module to assess understanding of the material.
* A final project will be assigned, which will involve applying number theory concepts to a real-world problem or cryptography.
**Final Project**
For the final project, students will choose one of the following options:
1. Implement RSA encryption using a programming language and test its security using number theory.
2. Write a program in your preferred programming language to generate a random prime number for use in cryptography.
3. Solve a cryptographic problem using number theory (e.g. factorization, discrete logarithm).
The final project will be presented in class and evaluated based on its originality, clarity, and technical accuracy.
**Assessment Breakdown**
* Quizzes and exams: 40%
* Homework assignments: 20%
* Final project: 30%
**Prerequisites**
* Basic knowledge of algebra and calculus
* Familiarity with programming languages (optional)
**Recommended Textbooks**
1. « A Course in Number Theory » by Henryk Iwaniec and Emmanuel Kowalski
2. « Number Theory: An Introduction to the Wonders of Cryptography » by Richard A. Mollin
3. « Cryptography: Theory and Practice » by Douglas Stinson
**Note**
This course outline is just a sample and may need to be adapted based on your specific teaching style, student needs, and institutional requirements. »