Prime Numbers

This book presents prime numbers in a beginner-friendly way, offering detailed proofs, definitions, examples, and exercises on key theorems and concepts. Each chapter includes exercises designed to reinforce and practice the concepts of prime numbers.

The book introduces prime and composite numbers, covering primality tests and the Sieve of Eratosthenes. It formalizes prime factorization as tuples and presents a straightforward method for finding the prime factorization of integers. The book also explains how to determine the GCD and LCM of a set of numbers using their prime factorizations.

Through detailed examples, we illustrate the Sieve of Eratosthenes for identifying prime numbers. We establish the infinitude of primes with several proofs, including Euclid's famous argument.

The book discusses Mersenne and Fermat primes, along with significant conjectures such as the twin prime conjecture, Goldbach's conjecture, and the Mersenne prime conjecture. We also cover primes in arithmetic progression and applications of Dirichlet's theorem.

Finally, we explore the distribution of primes using Bertrand's postulate, the prime-counting function, and the prime number theorem.

Here are some of the topics you will learn:

1. prime and composite numbers
2. primality test
3. sieve of Eratosthenes
4. prime factorization
5. infinitude of primes
6. Mersenne primes
7. Fermat primes
8. Goldbach conjecture
9. twin prime conjecture
10. prime number theorem
11. Bertrand's postulate (Chebyshev's theorem)