Lecture: Prime Numbers and Beyond

Introduction

• Presenter: Ravi Prakash
• Topic: Continuation of prime numbers discussion from Numbers 2.

Key Concepts about Prime Numbers

Unique Properties

• Even Prime Number: 2 is the only even prime number.
• Divisibility by 24: For any prime number p, ( p^2 - 1 ) is divisible by 24.
  • Proof: Let ( p = 6k \pm 1 ); upon squaring, ( p^2 - 1 ) is always divisible by 24.
• Digital Sum: The digital sum of a prime number can never be 3, 6, or 9.

Digital Sum Explained

• Definition: Add the digits of a number until a single digit is obtained.
• Example: For 949876, digital sum is 7.
• Importance: Used in algebra, geometry, and option elimination.

Identifying Prime Numbers

Quick Identification

• Calculate the square root of n.
• Check divisibility by all prime numbers less than or equal to the square root.
  • Example: 143 is not prime as it is divisible by 11.
  • Example: 731 is not prime as it is divisible by 17.

Logical Explanation

• Factor Pairs: Numbers can be expressed as products of factor pairs around their square root.

Methods to Check Prime Numbers

1. Form: Check if the number is of the form ( 6k \pm 1 ).
2. Digital Sum: Should not be 3, 6, or 9.
3. Divisibility: Check ( p^2 - 1 ) for divisibility by 24.
4. Divisibility by Primes: Check divisibility by primes before the square root.

Composite Numbers

• Definition: Natural numbers with 3 or more factors.
• Note: 1 is neither prime nor composite.

Examples & Questions

Prime or Composite

• Example 1: ( 3^{193} + 5 ) is composite (it's even).
• Example 2: ( 10001 ) is composite (divisible by 101).

Concepts of ( a^n \pm b^n )

• Odd n: ( a^n + b^n ) is divisible by ( a + b ).
• Even n: No definite conclusion for ( a^n + b^n ).
• Odd n: ( a^n - b^n ) is divisible by ( a - b ).
• Even n: ( a^n - b^n ) is divisible by both ( a - b ) and ( a + b ).

Application of Concepts

• Example 3: ( 2^{3007} + 1 ) is composite (divisible by 3).
• Example 4: 973 is composite (divisible by 7).

Conclusion

• Discussed how to determine if a number is prime or composite using various mathematical insights and properties.
• These understandings are crucial for solving questions related to prime numbers effectively.