Number Theory
For Beginners
1. Introduction to Numbers:
- Natural Numbers (N): These are the counting numbers 1, 2, 3, 4, and so on. They are used for basic counting.
- Whole Numbers (W): These include zero along with the natural numbers.
- Integers (Z): Integers consist of positive and negative whole numbers, including zero.
2. Primes and Composites:
- Prime Numbers: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Examples include 2, 3, 5, 7, 11, etc.
- Composite Numbers: Numbers that are not prime are composite. They can be factored into smaller natural numbers.
3. Divisibility:
- A number (a) is said to be divisible by another number (b) if there exists an integer (k) such that a = b * k.
- Divisibility Rules: For example, a number is divisible by 2 if its last digit is even, by 3 if the sum of its digits is divisible by 3, and so on.
4. Greatest Common Divisor (GCD) and Least Common Multiple (LCM):
- GCD: The greatest common divisor of two numbers is the largest number that divides both of them. For example, the GCD of 12 and 18 is 6.
- LCM: The least common multiple of two numbers is the smallest number that is a multiple of both. For example, the LCM of 4 and 6 is 12.
5. Euclidean Algorithm:
- The Euclidean Algorithm is a method for finding the GCD of two numbers. It involves repeated application of the division algorithm.
6. Fundamental Theorem of Arithmetic:
- Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors.
7. Modular Arithmetic:
- Modulus Operator: In modular arithmetic, numbers "wrap around" after reaching a certain value (the modulus). For example, in modulo 7, 10 ≡ 3 because 10 divided by 7 leaves a remainder of 3.
8. Congruences:
- If (a) and (b) have the same remainder when divided by (m), we say that (a) is congruent to (b) modulo (m) and write a ≡ b (mod m).
9. Fermat's Little Theorem:
- If (p) is a prime number and (a) is an integer not divisible by (p), then a^(p-1) ≡ 1 (mod p).
10. Number Patterns and Sequences:
- Exploring patterns in prime numbers, twin primes, triangular numbers, and other sequences.
11. Diophantine Equations:
- Diophantine equations involve finding integer solutions to polynomial equations. An example is ax + by = c.
12. RSA Algorithm (Optional):
- An application of number theory in cryptography, where the security relies on the difficulty of factoring large composite numbers.
Conclusion:
Number theory is a vast and rich field with applications in various areas of mathematics and computer science. This lesson provides a foundational understanding, but there are many more advanced topics and unsolved problems awaiting exploration. As you delve deeper into number theory, you'll uncover the beauty and complexity of this timeless mathematical discipline.