Podcast
Questions and Answers
What is the definition of a divides b (a | b) in number theory?
What is the definition of a divides b (a | b) in number theory?
What is a prime number in number theory?
What is a prime number in number theory?
What is the GCD of two or more integers?
What is the GCD of two or more integers?
What is the definition of two integers a and b being congruent modulo n (a ≡ b (mod n))?
What is the definition of two integers a and b being congruent modulo n (a ≡ b (mod n))?
Signup and view all the answers
What is a Diophantine equation?
What is a Diophantine equation?
Signup and view all the answers
What is Fermat's Little Theorem?
What is Fermat's Little Theorem?
Signup and view all the answers
What is a property of divisibility?
What is a property of divisibility?
Signup and view all the answers
What is a method for finding the GCD of two or more integers?
What is a method for finding the GCD of two or more integers?
Signup and view all the answers
Study Notes
Number Theory
Divisibility
- A positive integer a is said to divide another integer b (written as a | b) if there exists an integer k such that b = ka
- Properties of divisibility:
- If a | b and a | c, then a | (b + c)
- If a | b and b | c, then a | c
Prime Numbers
- A prime number is a positive integer that is divisible only by itself and 1
- Properties of prime numbers:
- There are an infinite number of prime numbers
- Every positive integer can be expressed as a product of prime numbers in a unique way (Fundamental Theorem of Arithmetic)
- Prime numbers are greater than 1
Greatest Common Divisor (GCD)
- The GCD of two or more integers is the largest integer that divides each of them without leaving a remainder
- Methods for finding GCD:
- Prime factorization
- Euclidean algorithm
Congruences
- Two integers a and b are said to be congruent modulo n (written as a ≡ b (mod n)) if they leave the same remainder when divided by n
- Properties of congruences:
- If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n)
- If a ≡ b (mod n), then a ≡ b + kn (mod n) for any integer k
Diophantine Equations
- A Diophantine equation is a polynomial equation in two or more variables with integer coefficients, for which only integer solutions are sought
- Linear Diophantine equations:
- ax + by = c, where a, b, and c are integers and x and y are variables
- Solutions exist if and only if c is a multiple of the GCD of a and b
Fermat's Little Theorem
- If p is a prime number, then a^p ≡ a (mod p) for any integer a
- Applications:
- Testing for primality
- Cryptography
Divisibility
- A positive integer a divides another integer b if there exists an integer k such that b = ka.
- Key properties include:
- If a divides both b and c, then it also divides their sum: a | (b + c).
- If a divides b and b divides c, then a divides c: a | c.
Prime Numbers
- A prime number is defined as a positive integer greater than 1 that is only divisible by itself and 1.
- Notable properties of primes:
- There are infinitely many prime numbers.
- Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers, known as the Fundamental Theorem of Arithmetic.
Greatest Common Divisor (GCD)
- The GCD of two or more integers is the largest integer that can divide each without leaving a remainder.
- Common methods to determine the GCD include:
- Prime factorization, which involves expressing each number as a product of primes.
- The Euclidean algorithm, based on repeated division.
Congruences
- Two integers a and b are congruent modulo n if they yield the same remainder when divided by n, denoted as a ≡ b (mod n).
- Congruences follow certain properties:
- If a ≡ b (mod n) and c ≡ d (mod n), then their sums are also congruent: a + c ≡ b + d (mod n).
- If a ≡ b (mod n), then for any integer k, a remains congruent to b + kn (mod n).
Diophantine Equations
- A Diophantine equation consists of polynomial equations with integer coefficients, specifically seeking integer solutions.
- For linear Diophantine equations of the form ax + by = c, solutions exist if and only if c is a multiple of the GCD of a and b.
Fermat's Little Theorem
- When p is a prime number, for any integer a, the relationship a^p ≡ a (mod p) holds.
- This theorem has practical applications in:
- Primality testing, offering a way to determine if a number is prime.
- Cryptography, particularly in algorithms that rely on number theory.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Learn about the fundamentals of number theory, including divisibility, prime numbers, and their properties. Understand how to determine if a number is prime and how divisibility works.
