1. Basic Concepts in Number Theory
a) Integers:
- The set of integers () includes positive and negative whole numbers as well as zero:
b) Divisibility:
- An integer divides another integer (denoted as ) if there exists an integer such that
c) Prime Numbers:
- A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. For example, are prime numbers.
d) Greatest Common Divisor (GCD):
- The GCD of two integers and is the largest number that divides both and .
- For example,
e) Least Common Multiple (LCM):
- The least common multiple of two integers and is the smallest positive integer divisible by both. For example, .
2. Prime Numbers and Factorization
a) Fundamental Theorem of Arithmetic:
- Every integer greater than 1 can be factored uniquely into a product of prime numbers. This is the basis of prime factorization. For example:
b) Sieve of Eratosthenes:
- A method to find all prime numbers less than a given . By eliminating multiples of each prime starting from 2, we identify the primes in a range.
3. Modular Arithmetic
a) Concept:
- Modular arithmetic involves numbers wrapping around after reaching a certain value, called the modulus. For example, since 12 leaves a remainder of 0 when divided by 12.
b) Basic Properties:
- Modular arithmetic obeys the basic rules of addition, subtraction, and multiplication. For example:
4. Linear Congruences
a) Definition:
- A linear congruence takes the form where and are integers. Solving for involves finding values of that satisfy the equation within the modular system.
b) Solving Linear Congruences:
- Solving these equations often requires using the Euclidean algorithm to find the multiplicative inverse of modulo .
5. Diophantine Equations
a) Definition:
- Diophantine equations are polynomial equations where integer solutions are sought. For example: where are integers and are unknowns.
b) Solutions:
- A solution to this equation exists if and only if divides .
6. Theorems in Number Theory
a) Fermat's Little Theorem:
- If is a prime and is not divisible by , then:
This theorem has applications in cryptography and computational mathematics.
b) Euler’s Theorem:
- For any integer coprime to : where is Euler’s totient function, counting the integers up to that are coprime to .
7. Cryptography and Number Theory
a) RSA Encryption:
- RSA encryption is a public-key cryptographic system based on the difficulty of factoring large integers. It relies on modular arithmetic and the properties of prime numbers.
b) Diffie-Hellman Key Exchange:
- This method uses modular exponentiation to enable two parties to securely exchange cryptographic keys.
8. Applications of Elementary Number Theory
a) Cryptography:
- Elementary number theory is central to encryption methods, such as RSA, which secure online communications.
b) Computer Algorithms:
- Many algorithms, like those for primality testing, are rooted in number theory. Modular arithmetic is also essential for hash functions in computer science.
c) Pure Mathematics:
- Number theory provides tools for exploring deep mathematical problems, including unsolved questions like the Goldbach conjecture or the Riemann hypothesis.
Some Examples
1. Divisibility and Prime Numbers
Example 1: Divisibility
- Determine whether (i.e., check if 18 divides 54).
- Solution: which is an integer. Therefore, .
Example 2: Prime Numbers
- Check if 29 is a prime number.
- Solution: To verify, we check divisibility by all prime numbers less than (i.e., 2, 3, 5). Since 29 is not divisible by any of them, 29 is prime.
2. Greatest Common Divisor (GCD)
Example 3: Using Euclidean Algorithm
- Find the GCD of 56 and 98 using the Euclidean algorithm.
- Solution: Apply the division algorithm: Since the remainder is 0, the GCD is the last non-zero remainder, .
3. Modular Arithmetic
Example 4: Modular Addition
- Compute
- Solution: First, add the numbers: Now, divide by 6 and take the remainder: Therefore, .
Example 5: Modular Multiplication
- Compute
- Solution: First, multiply the numbers: Now, divide by 7 and take the remainder:
Therefore,
- Solution: First, multiply the numbers: Now, divide by 7 and take the remainder:
Therefore,
4. Linear Diophantine Equation
Example 6: Solving a Diophantine Equation
- Solve for integer values of and .
- Solution: First, find the GCD of 15 and 21, which is 3 (since and ).
- Divide the equation by 3: Using the extended Euclidean algorithm, we solve for and : Working backwards: Thus, is a particular solution to the equation.
5. Fermat's Little Theorem
Example 7: Using Fermat’s Little Theorem
- Find .
- Solution: Fermat's Little Theorem states that for a prime , if is not divisible by then: Here, and Now, write Compute and
These examples illustrate key concepts of divisibility, modular arithmetic, GCD, Diophantine equations, and theorems like Fermat’s Little Theorem.