Number Theory in Mathematics

Number theory is a branch of mathematics that studies numbers, particularly whole numbers, and their properties and relationships. It explores patterns, structures, and the behaviors of numbers in different situations. Number theory deals with the following key concepts:

• Prime Numbers: Properties, distribution, and applications of prime numbers.
• Divisibility: Rules and relationships of numbers dividing each other.
• Greatest Common Divisor (GCD) and Least Common Multiple (LCM): Finding common factors and multiples.
• Modular Arithmetic: Remainders and "clock arithmetic."
• Number Patterns: Squares, cubes, and other numerical sequences.
• Congruences: Relationships between numbers in modular systems.

Number System

This section explains the different kinds of numbers used in mathematics, from natural to complex numbers, and various number systems like binary and decimal.

• Number System
  • Natural Numbers
  • Whole Numbers
  • Integers
  • Rational Numbers
  • Irrational Numbers
  • Real Numbers
  • Imaginary Numbers
  • Complex Numbers
• Types of Number System
  • Binary Number System
  • Octal Number System
  • Decimal Number System
  • Hexadecimal Number System

Basic Concepts

Here, you'll learn foundational number theory concepts like divisors, multiples, prime numbers, HCF, LCM, and modular arithmetic.

• Divisors
  • Divisibility Rules
• • Sum of Divisors
  • Number of Divisors
• Multiples
  • Multiplication Tables
  • HCF and LCM
• Prime Numbers
  • Prime Factorization
  • Sieve of Eratosthenes
  • Segmented Sieve
• Greatest Common Divisor (GCD)
  • Euclidean Algorithm
  • Extended Euclidean Algorithm
• Least Common Multiple (LCM)
• Modular Arithmetic
  • Modular Addition
  • Modular Multiplication: Modular Multiplicative Inverse
  • Modular Division
  • Modular Exponentiation
  • Fermat's Little Theorem
• Co-Prime Numbers
• Euler’s Totient Function

Advanced Concepts

Dive into deeper number theory topics such as the Chinese Remainder Theorem, Diophantine equations, and number-theoretic identities.

• Chinese Remainder Theorem
• Wilson’s Theorem
• Diophantine Equations
• Linear Diophantine Equations
• Pell’s Equation
• Bezout's Identity
• Mobius Function
• Mobius Inversion

Prime Distribution

Understand how prime numbers are spread across the number line and explore interesting patterns and famous conjectures related to them.

• Distribution of Primes
• Prime Number Theorem
• Goldbach's Conjecture
• Patterns in Primes
  • Twin Primes
  • Cousin Primes
• • Prime Triplets
  • Prime Quadruplets

Miscellaneous Topics of Number Theory

This section includes special and fascinating types of numbers and principles like Fibonacci numbers, perfect numbers, and the Pigeonhole Principle.

• Catalan Numbers
• Fibonacci Sequence
• Farey Sequences
• Pigeonhole Principle
• Perfect Numbers
• Deficient Numbers
• Abundant Numbers
• Amicable Numbers
• Automorphic Numbers
• Magic Numbers
• Triangular Numbers
• Tetrahedral Number
• Hexagonal Numbers
• Lucas Primes
• Hardy-Ramanujan Numbers

Programs for Number Theory

Practice number theory through programming with code examples that solve problems like GCD, LCM, primality testing, and more.

• Find the GCD of two number
• Find the LCM of two number
• Calculate the Factorial of a number
• Basic and Extended Euclidean algorithms
• Primality Test to check if a number is prime or not
• Primality Test to check if a number is prime or not using the Fermat Method
• Primality Test to check if a number is prime or not using Miller–Rabin
• Primality Test to check if a number is prime or not using Solovay-Strassen

Number Theory Books

This section lists useful books to help you learn and understand number theory better.

• Elementary Number Theory by David M. Burton
• An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright
• A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen
• Algebraic Number Theory by Jürgen Neukirch