Step-by-Step Instructions

1. 1.
   Enter a Number: Input any positive integer into the prime number checker. The calculator accepts numbers up to 999,999,999.
2. 2.
   Click "Analyze Number": The primality test calculator will instantly determine if your number is prime.
3. 3.
   View Results: Get detailed prime factorization, all factors, neighboring primes, and performance metrics.
4. 4.
   Analyze Patterns: Explore prime number patterns and properties with the comprehensive results.

Calculator Features

Primality Test Algorithm

Our prime number checker uses an optimized trial division algorithm for efficient primality testing:

function isPrime(n):
  if n ≤ 1: return false
  if n ≤ 3: return true
  if n % 2 == 0 or n % 3 == 0: return false
  
  for i from 5 to √n, step 6:
    if n % i == 0 or n % (i + 2) == 0:
      return false
  return true

Prime Factorization Method

The prime factor calculator uses trial division to find all prime factors:

function primeFactorization(n):
  factors = []
  
  // Handle factor 2 separately
  while n % 2 == 0:
    factors.add(2)
    n = n / 2
  
  // Handle odd factors
  for i from 3 to √n, step 2:
    while n % i == 0:
      factors.add(i)
      n = n / i
  
  // If remaining n is prime
  if n > 2:
    factors.add(n)
  
  return factors

Time Complexity

Common Prime Number Examples

Prime Factorization Examples

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. Examples include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on. Use our prime number checker to test any number.

How to check if a number is prime?

To check if a number is prime, you can use our primality test calculator. The algorithm tests divisibility by all numbers up to the square root of the number. If no divisors are found, the number is prime.

Is 1 a prime number?

No, 1 is not considered a prime number. By definition, a prime number must have exactly two distinct positive divisors: 1 and itself. Since 1 only has one divisor (itself), it's not prime.

What is the largest prime number?

There is no largest prime number - there are infinitely many primes. However, the largest known prime number (as of 2024) is 2^82,589,933 - 1, which has over 24 million digits.

How does the prime factor calculator work?

Our prime factor calculator uses trial division to find all prime factors of a number. It systematically tests divisibility by prime numbers and builds the complete prime factorization.

What is the difference between prime and composite numbers?

Prime numbers have exactly two divisors (1 and themselves), while composite numbers have more than two divisors. For example, 7 is prime (divisors: 1, 7) but 6 is composite (divisors: 1, 2, 3, 6).

Why are prime numbers important in cryptography?

Prime numbers are fundamental to modern cryptography, particularly in RSA encryption. The difficulty of factoring large prime numbers provides security for digital communications, online banking, and secure data transmission.

What is a primality test?

A primality test is an algorithm used to determine if a given number is prime. Our primality test calculator uses efficient algorithms to quickly determine primality for numbers up to 999,999,999.