Questions and Answers
What is the definition of a divides b (a | b) in number theory?
What is a prime number in number theory?
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))?
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What is a Diophantine equation?
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What is Fermat's Little Theorem?
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What is a property of divisibility?
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What is a method for finding the GCD of two or more integers?
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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.
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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.
