Introduction to Number Theory
Number theory is one of the oldest and most fascinating branches of mathematics, often referred to as the "Queen of Mathematics." It primarily deals with the properties and relationships of integers—the whole numbers we use every day—and has evolved from an abstract field to one with profound real-world applications.
At its core, number theory explores the fundamental properties of numbers and the patterns they form. While historically pursued for its intrinsic mathematical beauty, number theory now forms the backbone of modern cryptography, secure communications, digital signatures, cryptocurrency, computer algorithms, and data security systems that protect everything from your social media accounts to global financial transactions.
Divisibility and Prime Numbers
A fundamental concept in number theory is divisibility. We say an integer $a$ divides an integer $b$ (written as $a|b$) if there exists an integer $c$ such that $b = a \cdot c$. When $a$ divides $b$, we call $a$ a divisor or factor of $b$.
Example:
Consider the number 24:
- The divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24
- We can write: $1|24$, $2|24$, $3|24$, etc.
- For instance, $8|24$ because $24 = 8 \cdot 3$
- However, $5 \nmid 24$ (5 does not divide 24) because there is no integer $c$ where $24 = 5 \cdot c$
Prime numbers are integers greater than 1 that have exactly two distinct positive divisors: 1 and themselves. For example, 2, 3, 5, 7, 11, and 13 are prime numbers. All other integers greater than 1 are called composite numbers.
