Prime Number Basics
A Prime Number is a core concept in number theory and the foundation of modern cryptography (e.g., RSA). It is a natural number greater than $1$ that has no positive divisors other than $1$ and itself.
1. Definition and Characteristics
① Definitions
- Prime Number: A natural number greater than $1$ that is only divisible by $1$ and itself.
- Examples: $2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$
- Composite Number: A natural number greater than $1$ that has at least one divisor other than $1$ and itself.
- Examples: $4, 6, 8, 9, 10, 12, \dots$
- The number 1 is neither prime nor composite. It acts as the unit of multiplication.
② Fundamental Property: Prime Factorization
According to the Fundamental Theorem of Arithmetic, every integer greater than $1$ can be uniquely represented as a product of prime numbers.
- Example: $12 = 2 \times 2 \times 3$, $91 = 7 \times 13$
[Image of Prime Factorization Tree]
2. Significance of Prime Numbers
① Core of Number Theory
Primes are the basic "building blocks" for all natural numbers and play a fundamental role in every area of mathematics dealing with integers.
② Core of Modern Cryptography
Primes are essential for ensuring the confidentiality of modern security systems.
- RSA Encryption: RSA works by multiplying two extremely large prime numbers to create a public key. The security relies on the fact that while it's easy to multiply them, it's computationally infeasible (too time-consuming) to reverse the process and find the two original primes (prime factorization).
- Security Principle: The algorithm exploits the asymmetric difficulty: easy to find large primes, hard to factor their product.
③ Distribution of Primes
Primes are infinite, yet their pattern of appearance is irregular and hard to predict, making them vital in random number generation and cryptographic research.