Number Theory

数論

日本語版

About This Series

Number theory is the branch of mathematics that studies the properties of integers. Often called "the queen of mathematics," it is one of the oldest areas of research yet remains an active field with many open problems today.

This series progresses from the basic properties of integers through the distribution of primes, the theory of congruences, and on to algebraic and analytic number theory. Applications to cryptography and computer science are discussed throughout.

Learning by Level

Introduction

High-school level

  • Properties of integers, primes, and factorization
  • The Euclidean algorithm
  • Remainders, classification, and proof techniques
  • Positional notation (binary, hexadecimal)

Elementary

Undergraduate 1st–2nd year — Theory

  • Basics and arithmetic of congruences
  • Fermat's and Euler's theorems
  • Chinese Remainder Theorem, primitive roots
  • Mathematical foundations of RSA

Intermediate

Undergraduate 3rd–4th year — Applications

  • Applications of congruences, Euler's function, CRT
  • The discrete logarithm problem
  • Miller–Rabin primality test
  • Binary GCD, elliptic curves

Advanced

Graduate level

  • Modular forms
  • Fermat's Last Theorem
  • Arithmetic geometry

Learning Path

Intro High school Elementary Undergrad 1–2 Intermediate Undergrad 3–4 Advanced Graduate Intro: divisibility, primes, Euclidean alg., remainders, proofs, bases Elementary: congruences, Fermat/Euler, CRT, primitive roots (theory) Intermediate: discrete log, Miller–Rabin, elliptic curves (applications) Advanced: modular forms, FLT, arithmetic geometry

Main Topics

Divisibility

Divisors and multiples, prime factorization, GCD and LCM.

Primes

The infinitude of primes, the Prime Number Theorem, and open problems such as the twin prime conjecture.

Modular Arithmetic

Computation in the world of remainders; Fermat's and Euler's theorems; applications to RSA cryptography.

Diophantine Equations

Equations seeking integer solutions: Pythagorean triples and Fermat's Last Theorem.

Special Topics

In-depth topics that cut across levels.

Integer Factorization Algorithms

A central topic in computational number theory

  • Trial division, Fermat's method
  • Pollard's rho and p−1 methods
  • Elliptic Curve Method (ECM)
  • Quadratic sieve, number field sieve
  • Connection to RSA cryptography

Fields of Application

  • Cryptography: RSA, elliptic-curve cryptography, and other public-key systems
  • Coding theory: Construction of error-correcting codes
  • Computer science: Primality testing and factorization algorithms
  • Physics: Modular forms in string theory