Number Theory – The Study of Integers and Their Properties
Number theory is the branch of pure mathematics devoted entirely to the whole numbers – their divisibility, their patterns, and the surprisingly deep structure hiding inside something as simple as counting. Carl Friedrich Gauss, one of history's greatest mathematicians, called it “the queen of mathematics,” and mathematicians have spent over two thousand years proving that he was right: some of the field's most famous questions are simple enough to explain to a child, yet took centuries – or remain unsolved – to answer.
The subject's roots trace back to Euclid of Alexandria, whose Elements (c. 300 BCE) devoted three of its thirteen books to number theory, proving that there are infinitely many primes and describing the algorithm for finding the greatest common divisor that still bears his name today. About five centuries later, Diophantus of Alexandria wrote the Arithmetica, a collection of problems asking for whole-number solutions to equations – a type of problem now called a “Diophantine equation” in his honour. In the 17th century the French lawyer and amateur mathematician Pierre de Fermat scribbled a note in the margin of his copy of the Arithmetica claiming to have a proof that xn + yn = zn has no whole-number solutions for n > 2 – a claim, now known as Fermat's Last Theorem, that was not proved until 1994, 358 years later, by the British mathematician Andrew Wiles. In 1801, Gauss published his Disquisitiones Arithmeticae, which organised the whole field around a single powerful idea – congruence – and effectively founded number theory as a modern, systematic discipline.
What Is Number Theory?
Number theory studies the properties and relationships of integers (whole numbers): which numbers divide which, how remainders behave, which numbers are prime, and how equations behave when only whole-number solutions are allowed. It began as pure curiosity about numbers themselves, with no practical goal – yet it now sits at the heart of the encryption that protects almost every online transaction.
The Main Branches of Number Theory
| Branch | What It Studies | Examples |
|---|---|---|
| Elementary Number Theory | Divisibility, primes, and congruences using elementary methods | Divisibility rules, modular arithmetic, GCD/LCM |
| Analytic Number Theory | Uses calculus and analysis to study whole numbers | The distribution of prime numbers |
| Algebraic Number Theory | Extends number theory using algebraic structures | Rings, fields, algebraic integers |
| Computational Number Theory | Designs fast algorithms for number-theoretic problems | Primality testing, integer factorisation, cryptography |
Core Building Blocks
Every deeper idea in this section of MathsFamily builds on a handful of core concepts:
| Concept | Simple Description |
|---|---|
| Divisibility | Whether one integer divides another with no remainder. |
| Congruence | Two numbers that leave the same remainder when divided by n. |
| Greatest Common Divisor | The largest number that divides two or more integers exactly. |
| Prime Number | A number greater than 1 whose only factors are 1 and itself. |
Why Number Theory Matters
- Cryptography: RSA encryption, which secures much of the internet, relies directly on the difficulty of factoring large numbers into primes.
- Computer Science: Hash functions, checksums, and error-correcting codes all use modular arithmetic.
- Calendars and Scheduling: Working out repeating cycles (days of the week, leap years) is modular arithmetic in disguise.
- Competition Mathematics: Number theory is a core pillar of maths olympiads worldwide.
- Pure Curiosity: Famous open problems – like the Riemann Hypothesis and the Goldbach Conjecture – remain unsolved after centuries, and carry million-dollar prizes for anyone who cracks them.
What You Will Learn
This Number Theory section on MathsFamily builds from modular arithmetic and congruences through Diophantine equations, Fermat's Little Theorem, Euler's totient function, perfect and amicable numbers, and finally to the number theory behind modern cryptography. Each topic builds on the last, so start here and work through in order.
Key Takeaways
- Number theory is the pure study of integers – divisibility, primes, and their relationships.
- It has four main branches: elementary, analytic, algebraic, and computational.
- Founders include Euclid, Diophantus, Fermat, and Gauss, spanning over two thousand years.
- Despite its ancient, abstract origins, number theory now underpins modern internet security.