Number Theory Study Notes

Study Notes

Number Theory Study Notes

Definition:
- A branch of pure mathematics focused on the properties and relationships of numbers, particularly integers.
Key Concepts:
- Integers: Whole numbers that can be positive, negative, or zero.
- Prime Numbers: Natural numbers greater than 1 that have no positive divisors other than 1 and themselves (e.g., 2, 3, 5, 7).
- Composite Numbers: Natural numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9).
- Greatest Common Divisor (GCD): Largest integer that divides two or more integers without leaving a remainder.
- Least Common Multiple (LCM): Smallest integer that is a multiple of two or more integers.
Divisibility Rules:
- A number is divisible by:
  - 2 if its last digit is even.
  - 3 if the sum of its digits is divisible by 3.
  - 5 if its last digit is 0 or 5.
  - 10 if it ends in 0.
Fundamental Theorem of Arithmetic:
- Every integer greater than 1 can be uniquely factored into prime numbers.
Modular Arithmetic:
- A system of arithmetic for integers where numbers "wrap around" after reaching a certain value (the modulus).
- Notation: a ≡ b (mod n) means a and b have the same remainder when divided by n.
Congruences:
- If a ≡ b (mod n), then a and b are congruent modulo n.
- Properties include:
  - If a ≡ b and c ≡ d, then a + c ≡ b + d and a * c ≡ b * d.
Diophantine Equations:
- Polynomial equations that seek integer solutions (e.g., ax + by = c).
Fermat's Last Theorem:
- States there are no three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.
Applications:
- Cryptography: Uses properties of number theory to secure data through algorithms such as RSA.
- Computer Science: Algorithms for primality testing, factoring, etc.
Important Theorists:
- Euclid: Known for his work in geometry but also made significant contributions to number theory.
- Pierre de Fermat: Famous for Fermat's Last Theorem.
- Carl Friedrich Gauss: Made foundational contributions, including the Law of Quadratic Reciprocity.
Further Topics:
- Quadratic residues and non-residues.
- The distribution of prime numbers (e.g., Prime Number Theorem).
- Continued fractions and their applications.

This structure provides a concise overview of number theory, touching upon essential concepts and applications relevant to the field.

Definition

Branch of pure mathematics concentrating on the properties and relationships of numbers, especially integers.

Key Concepts

Integers: Whole numbers, which can be positive, negative, or zero.
Prime Numbers: Natural numbers greater than 1 with no divisors other than 1 and themselves (e.g., 2, 3, 5, 7).
Composite Numbers: Natural numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9).
Greatest Common Divisor (GCD): The largest integer that divides two or more integers without a remainder.
Least Common Multiple (LCM): The smallest integer that is a multiple of two or more integers.

Divisibility Rules

A number is divisible by:
- 2 if its last digit is even.
- 3 if the sum of its digits is divisible by 3.
- 5 if its last digit is 0 or 5.
- 10 if it ends in 0.

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be uniquely expressed as a product of prime numbers.

Modular Arithmetic

An arithmetic system for integers where numbers "wrap around" after reaching a specified value, called the modulus.
Notation a ≡ b (mod n) indicates that a and b yield the same remainder when divided by n.

Congruences

If a ≡ b (mod n), then a and b are congruent modulo n.
Key properties include:
- If a ≡ b and c ≡ d, then a + c ≡ b + d and a × c ≡ b × d.

Diophantine Equations

Polynomial equations that seek integer solutions, exemplified by the equation ax + by = c.

Fermat's Last Theorem

Asserts that no three positive integers a, b, and c can satisfy a^n + b^n = c^n for any integer n greater than 2.

Applications

Cryptography: Utilizes number theory properties to secure data through algorithms like RSA.
Computer Science: Involves algorithms for primality testing, factoring, and other computations.

Important Theorists

Euclid: Renowned for geometry but also influential in number theory.
Pierre de Fermat: Best known for Fermat's Last Theorem.
Carl Friedrich Gauss: Made significant contributions, including the Law of Quadratic Reciprocity.

Further Topics

Exploration of quadratic residues and non-residues.
Study of prime number distribution, including the Prime Number Theorem.
Investigations into continued fractions and their applications.

Description

Explore the fundamental concepts of number theory, a branch of pure mathematics. This quiz covers key topics such as integers, prime and composite numbers, divisibility rules, and the foundational theorem of arithmetic. Perfect for students looking to deepen their understanding of number properties.