1. What is a Prime Number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are the building blocks of all natural numbers through multiplication.

2. How Does the Prime Checker Work?

The calculator uses optimized prime checking algorithm:

Algorithm Steps:

• Check if number ≤ 1 → not prime
• Check if number ≤ 3 → prime
• Check divisibility by 2 and 3
• Check divisors of form 6k±1 up to √n

Explanation: This method efficiently checks for prime numbers by reducing the number of divisions needed, making it much faster than checking all numbers up to n.

3. Importance of Prime Numbers

Details: Prime numbers are fundamental in mathematics and have crucial applications in cryptography, computer science, and number theory. They form the basis for modern encryption algorithms like RSA.

4. Using the Calculator

Tips: Enter any integer greater than 1. The calculator will determine if it's prime or composite. For very large numbers, the calculation may take longer.

5. Frequently Asked Questions (FAQ)

Q1: Is 1 a prime number?
A: No, 1 is not considered a prime number because it has only one positive divisor (itself), while prime numbers must have exactly two distinct positive divisors.

Q2: What is the largest known prime number?
A: As of 2024, the largest known prime is 2^82,589,933 − 1, a number with 24,862,048 digits.

Q3: Are there infinite prime numbers?
A: Yes, Euclid proved around 300 BC that there are infinitely many prime numbers.

Q4: What are twin primes?
A: Twin primes are pairs of primes that differ by 2, such as (3,5), (5,7), (11,13). It's unknown whether there are infinitely many twin primes.

Q5: Why check only up to √n?
A: If n has a divisor greater than √n, it must have a corresponding divisor less than √n, so checking up to √n is sufficient.