Overview

Number Theory is often called the “Queen of Mathematics” (Gauss). It is the study of the properties of whole numbers (integers), particularly prime numbers.

Core Idea

The core idea is investigating the atomic structure of numbers. Prime numbers are the “atoms” of arithmetic; every integer is uniquely built from them. Number theory asks how these atoms are distributed and how they behave.

Formal Definition

It studies the set of integers $\mathbb{Z}$ and subsets like natural numbers $\mathbb{N}$. Key concepts include:

  • Divisibility: $a$ divides $b$ ($a|b$).
  • Congruence: $a \equiv b \pmod n$ (Modular Arithmetic).
  • Primality: Numbers with exactly two factors.

Intuition

It’s the math of clocks and secrets.

  • Clock Math: $10 + 5 = 3$ (on a 12-hour clock). This is modular arithmetic.
  • Secrets: Multiplying two huge primes is easy; factoring the result back into primes is incredibly hard. This asymmetry protects the internet (RSA encryption).

Examples

  • Fermat’s Last Theorem: $x^n + y^n = z^n$ has no integer solutions for $n > 2$. (Proved by Wiles in 1994).
  • Goldbach Conjecture: Every even integer greater than 2 is the sum of two primes. (Still unproven).
  • Twin Prime Conjecture: There are infinitely many pairs of primes that differ by 2 (e.g., 11, 13).

Common Misconceptions

  • Misconception: It’s useless.
    • Correction: For centuries it was considered “pure” math with no application. Now, it is the foundation of digital security (Cryptography).
  • Misconception: We know everything about numbers.
    • Correction: Simple questions about primes remain some of the hardest unsolved problems in math (e.g., Riemann Hypothesis).
  • Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself.
  • Cryptography: The practice and study of techniques for secure communication.
  • Algorithm: Procedures for calculation (e.g., Euclidean algorithm for GCD).

Applications

  • Cryptography: RSA, Elliptic Curve Cryptography.
  • Coding Theory: Error correction codes.
  • Random Number Generation: Essential for simulation and gaming.

Criticism and Limitations

  • Difficulty: Problems are easy to state but notoriously difficult to solve, often requiring tools from completely different fields (like analysis or geometry).

Further Reading

  • An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright
  • The Music of the Primes by Marcus du Sautoy