An Introduction to Number Theory

Number theory is one of the oldest branches of mathematics, dealing with the properties of integers. Despite dealing with objects as simple as whole numbers, number theory contains some of the deepest and most difficult problems in all of mathematics.

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Every integer greater than 1 can be uniquely factored into primes (the Fundamental Theorem of Arithmetic). This uniqueness is what makes prime numbers so important.

GCD and LCM

The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both. The Euclidean algorithm finds it efficiently: repeatedly replace the larger number with the remainder of dividing it by the smaller. The Least Common Multiple (LCM) is the smallest number both divide into. The relationship GCD(a,b) × LCM(a,b) = a × b always holds.

Use our GCD and LCM Calculator and Prime Number Checker to explore these concepts.

Modular Arithmetic

Modular arithmetic is often called clock arithmetic. If it is 10 AM now and you add 4 hours, it is 2 PM, not 14. In modular arithmetic terms: 10 + 4 ≡ 2 (mod 12). This system is fundamental to cryptography, computer science, and many areas of pure mathematics.

Fibonacci Sequence

The Fibonacci sequence starts with 0, 1 and each subsequent term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... This sequence appears in nature (spiral arrangements of seeds, branching patterns), and the ratio of consecutive terms approaches the golden ratio φ = (1+√5)/2.