1. What Are Prime Numbers?

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. They are the fundamental building blocks of number theory and play a crucial role in mathematics and computer science.

2. How Does The Prime Test Work?

The calculator uses the trial division algorithm:

Where:

• \( n \) — The number being tested
• \( \sqrt{n} \) — Square root of n (upper limit for testing)
• \( n \mod i \) — Remainder when n is divided by i

Explanation: We only need to test divisors up to the square root of n because if n has a divisor greater than √n, it must also have a corresponding divisor less than √n.

3. Importance Of Prime Numbers

Details: Prime numbers are essential in cryptography (RSA encryption), computer science algorithms, number theory research, and are fundamental to modern digital security systems.

4. Using The Calculator

Tips: Enter any integer greater than 1. The calculator will test all possible divisors up to the square root of your number and determine if it's prime. Larger numbers will take slightly longer to compute.

5. Frequently Asked Questions (FAQ)

Q1: Why test only up to the square root?
A: If a number n has a divisor d greater than √n, then n/d must be a divisor less than √n. Testing up to √n covers all possible factor pairs.

Q2: What is the smallest prime number?
A: 2 is the smallest and only even prime number. All other primes are odd numbers greater than 2.

Q3: Are there infinite prime numbers?
A: Yes, Euclid proved over 2000 years ago that there are infinitely many prime numbers.

Q4: What are some common prime numbers?
A: Common small primes include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The largest known prime has millions of digits.

Q5: Why are primes important in cryptography?
A: The difficulty of factoring large numbers into their prime factors forms the basis of RSA encryption and many other cryptographic systems.