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 the fundamental theorem of arithmetic.

2. How Does the Calculator Work?

The calculator uses optimized trial division algorithm:

Algorithm Steps:

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

Next Prime Algorithm: Incrementally check subsequent numbers until a prime is found.

3. Importance of Prime Numbers

Details: Prime numbers are fundamental in mathematics and computer science. They are essential for cryptography (RSA encryption), number theory, and computational algorithms. Understanding primes helps in factorization problems and security systems.

4. Using the Calculator

Tips: Enter any positive integer to check if it's prime and find the next prime number. The calculator efficiently handles numbers up to reasonable computational limits.

5. Frequently Asked Questions (FAQ)

Q1: What is the smallest prime number?
A: The smallest prime number is 2, which is also the only even prime number.

Q2: Are there infinitely many prime numbers?
A: Yes, Euclid proved around 300 BC that there are infinitely many prime numbers.

Q3: What is a composite number?
A: A composite number is a positive integer that has at least one divisor other than 1 and itself.

Q4: How are prime numbers used in cryptography?
A: Large prime numbers are used in public-key cryptography systems like RSA, where the difficulty of factoring large numbers ensures security.

Q5: What is the largest known prime number?
A: As of 2024, the largest known prime is 2^82,589,933 − 1, a Mersenne prime with 24,862,048 digits.