Number Theory

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.

Some Applications of Number Theory in Computer Science:

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.
• GCD and 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

The Number System explains different kinds of numbers used in mathematics, from natural to complex numbers, and various number systems like binary and decimal.

• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
• Imaginary Numbers
• Complex Numbers

The four common types of number systems are:

• Binary Number System
• Octal Number System
• Decimal Number System
• Hexadecimal Number System

Basic Concepts

Foundational tools of Number Theory:

• Divisors & Multiples
• Divisibility Rules
• Sum of Divisors
• Number of Divisors
• HCF and LCM
• Prime Numbers
• Modular Arithmetic
• Fermat's Little Theorem
• Euler’s Totient Function
• Co-Prime Numbers

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
• Theorems & Conjectures
• 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

Number Theory For Programmers

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

• Find the GCD of two numbers
• Find the LCM of two numbers
• Calculate the Factorial of a number
• Basic and Extended Euclidean algorithms
• Primality Test: [Fermat Method] | [Miller–Rabin] | [Solovay-Strassen]