Prime Number Checker

Test if an integer up to 10¹² is prime, composite, or a multiplicative unit, list all positive divisors, find prime factorizations, and locate adjacent primes.

Category: Math

How to Use Prime Number Checker

  1. Enter any whole integer up to 1,000,000,000,000 (10¹²) in the 'Integer to Test' field.
  2. Click Calculate to run 6k ± 1 trial division primality verification.
  3. Inspect the classification (Prime, Composite, or Neither for ≤ 1), exponential prime factorization, divisor count, and nearest previous/next primes.

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Frequently Asked Questions

What is a prime number vs a composite number?

A prime number is a positive integer strictly greater than 1 that has exactly two positive divisors: 1 and itself (such as 2, 3, 5, 7, 11, 13, 997). A composite number is a positive integer greater than 1 that has three or more positive divisors (such as 4, 6, 8, 9, 12).

Why is the number 1 neither prime nor composite?

By definition, a prime number must have exactly two distinct positive divisors, whereas 1 has only one positive divisor (1 itself). Mathematically, 1 is classified as the multiplicative unit. If 1 were considered prime, the Fundamental Theorem of Arithmetic (unique prime factorization) would fail because any number could be factored with arbitrarily many 1s.

Can 0 or negative numbers be prime?

No. Prime and composite classifications apply strictly to positive integers greater than 1. Zero has infinitely many divisors (every non-zero integer divides 0), and negative integers are outside the domain of standard positive primes.

How fast is the 6k ± 1 trial division primality check?

Every prime greater than 3 can be written in the form 6k − 1 or 6k + 1. After checking divisibility by 2 and 3, our algorithm tests only candidate divisors of the form 6k ± 1 up to √n, allowing instant in-browser primality verification and prime factorization for integers up to 1,000,000,000,000 (10¹²).

How is the total number of positive divisors calculated from prime factors?

If an integer n has prime factorization p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, the divisor function d(n) equals the product of each exponent plus one: (a₁ + 1)(a₂ + 1)...(aₖ + 1). For example, 60 = 2² × 3¹ × 5¹ has (2 + 1)(1 + 1)(1 + 1) = 12 positive divisors.