Prime Number Calculator
Check if any number is prime or composite with detailed mathematical analysis
Check Prime Number
Enter any positive integer to check if it's prime or composite
Example Prime Tests
Prime Number: 17
Factors: 1, 17
Trial Division: Test up to √17 ≈ 4.1
17 ÷ 2 = 8.5 (not divisible)
17 ÷ 3 = 5.67 (not divisible)
Result: Prime ✓
Composite Number: 15
Factors: 1, 3, 5, 15
Trial Division: Test up to √15 ≈ 3.9
15 ÷ 2 = 7.5 (not divisible)
15 ÷ 3 = 5 (divisible!)
Result: Composite (15 = 3 × 5)
Prime Number Facts
The Odd Prime
2 is the only even prime number
Infinite Primes
Euclid proved there are infinitely many primes
Special Case
1 is neither prime nor composite
First 25 Prime Numbers
How It Works
Uses trial division algorithm
Tests divisibility up to √n
Optimized for efficiency
Shows step-by-step analysis
Understanding Prime Numbers
What is a Prime Number?
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. Prime numbers are the building blocks of all natural numbers through the Fundamental Theorem of Arithmetic.
Key Properties
- •Must be greater than 1
- •Has exactly two distinct factors
- •Cannot be formed by multiplying smaller natural numbers
- •2 is the only even prime number
Trial Division Algorithm
Step 1: Check if n ≤ 1 (not prime)
Step 2: Check if n = 2 or n = 3 (prime)
Step 3: Check if n is divisible by 2 or 3
Step 4: Test divisors from 5 to √n
Step 5: If no divisors found, n is prime
Why √n Limit?
If n has a factor greater than √n, it must also have a corresponding factor less than √n. So we only need to check up to √n to find all possible factors.
Applications of Prime Numbers
Cryptography
RSA encryption relies on the difficulty of factoring large numbers that are products of two primes.
Number Theory
Primes are fundamental in number theory and help understand the structure of integers.
Computer Science
Used in algorithms, hash functions, and pseudo-random number generation.