1. What is a Prime Number?

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number has exactly two distinct positive divisors: 1 and itself.

2. How Does the Prime Checker Work?

The calculator uses the trial division method:

Where:

• \( n \) — The number to test for primality
• \( \sqrt{n} \) — Square root of n (upper bound for divisor checking)

Explanation: If no divisors are found in the range [2, √n], then n is prime. This is because any composite number must have a factor less than or equal to its square root.

3. Importance of Prime Numbers

Details: Prime numbers are fundamental in mathematics and have crucial applications in cryptography, computer science, number theory, and security systems like RSA encryption.

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 are the first few prime numbers?
A: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Q3: Why check only up to the square root?
A: If n has a divisor greater than √n, then it must have a corresponding divisor less than √n. Checking beyond √n is redundant.

Q4: Are there infinite prime numbers?
A: Yes, Euclid proved around 300 BC that there are infinitely many prime numbers.

Q5: 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.