Optimized √n Check
Only checks divisors up to √n since if n = a × b and a > √n, then b < √n. Uses 6k±1 optimization.
Basic Trial Division
Checks all odd numbers from 3 to n-1. Slower but demonstrates the fundamental approach.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number can only be divided evenly by 1 and the number itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
Prime numbers are fundamental to number theory and have important applications in cryptography, computer science, and mathematics. The number 2 is the only even prime number, as all other even numbers are divisible by 2. Numbers greater than 1 that are not prime are called composite numbers.
The most straightforward way to test if a number n is prime is trial division: check if n is divisible by any integer from 2 to n-1. However, this can be optimized significantly by only checking up to √n, since if n = a × b and both a and b were greater than √n, then a × b would be greater than n.
Further optimization uses the fact that all primes greater than 3 are of the form 6k±1. This means we only need to check divisibility by 2, 3, and numbers of the form 6k±1 up to √n. For very large numbers, probabilistic tests like Miller-Rabin are used.
This prime number checker uses standard mathematical tests optimized for performance. For extremely large numbers (beyond the calculator's limit), advanced probabilistic methods like Miller-Rabin or AKS primality tests may be required for efficient verification.