Contextualization

First, let's briefly dive into the world of prime and composite numbers. The concept of prime and composite numbers is essential to understand numerous mathematical problems and is the basis of various branches of science, including cryptography, a technique used to protect information on the internet.

Prime numbers are those that have only two divisors, the number 1 and itself. They are fundamental in mathematics because every natural number greater than 1 is either prime or can be formed by the multiplication of prime numbers. This leads us to the concept of composite numbers, which are those that have more than two divisors.

Introduction

Prime and composite numbers are the foundation of arithmetic, one of the oldest and most fundamental branches of mathematics, dealing with numbers and numerical operations. The ability to differentiate and deal with these two types of numbers is essential for many areas of science and engineering.

In this project, you will deeply explore the fascinating universe of prime and composite numbers, not only learning to differentiate and manipulate them but also understanding their crucial role in various real-world contexts, from digital security to quantum physics.

Prime numbers have important practical applications in cryptography, which is the science of encoding information so that only authorized people can read it. The security of many online systems, such as banks and online shopping, depends on the difficulty of factoring large composite numbers into their primes.

Practical Activity

Activity Title: "Primes, Composites, and Cryptography: A Mathematical Investigation"

Project Objective:

This project aims to allow students to deepen their knowledge of prime and composite numbers, as well as the practical application of these concepts in cryptography. They will have to research the theory of these numbers and carry out a practical experiment involving cryptography, demonstrating their ability to apply theoretical knowledge in real-world situations.

Project Duration: Over 12 hours per student. Group Size: 3 to 5 students.

Detailed Project Description:

The project will be divided into two main parts: research on prime and composite numbers and their application in a cryptography experiment.

In the first part, students should research and study in-depth the concepts of prime and composite numbers, the difference between them, how to factor composite numbers into primes, as well as the importance of these numbers for various areas of knowledge.

In the second part of the project, students should apply this knowledge in a cryptography practice. They will have to create a simple encryption system using prime numbers and perform the encryption and decryption of some messages.

Required Materials:

• Books and various online resources for research on prime numbers, composite numbers, and cryptography.
• Paper and pen for notes and calculations.
• Computer with internet access for research and execution of the cryptography experiment.

Detailed Step-by-Step:

1. Theoretical Research: Groups should start by conducting comprehensive research on prime and composite numbers. The study should include the definition of each, how to identify a prime number, and how to factor composite numbers into primes.

2. Cryptography: With the knowledge acquired in the first part of the project, students should research the relationship between prime numbers and cryptography. They should understand how prime numbers are used in cryptography algorithms and why it is difficult to factor large composite numbers.

3. Practical Experiment: Students will create a simple encryption system, where they will use prime numbers to encrypt and decrypt messages. They should exchange messages with each other through this system and verify if the receiver can decrypt the message correctly.

4. Final Report: Each group must produce a detailed report of the project. The report should explain the theories studied, the practical experiment carried out, and the results obtained.

Project Deliverables:

The final project deliverables will be:

1. Written Report: The report should have sections on Introduction, Development, Conclusions, and Bibliography. Students should explain the theory studied, the methodology used, present and discuss the results obtained in their practical experiment. The bibliography used in the project should be indicated.

2. Practical Experiment: The encryption system created by the students, as well as the encrypted and decrypted messages, should be included in the final report.

The report and the results of the practical experiment should demonstrate that students were able to acquire technical skills in recognizing and differentiating prime and composite numbers, factoring composite numbers into primes, and applying this knowledge to real-world problems. Additionally, the teamwork required for this project will help students develop socio-emotional skills such as time management, communication, problem-solving, creative thinking, and proactivity.