Prime Number Checker
Test integers for primality and detect factors. Uses deterministic trial division optimized via 6k ± 1 wheel factorization to verify whether an integer has strictly two positive divisors (1 and itself).
Enter a positive whole number to evaluate
Whether the number is prime, composite, or neither
How to Use This Calculator
- Enter the positive integer you wish to test.
- View the clear Prime or Composite determination instantly.
- Review discovered divisors and the trial division boundary explanation.
Formula & Calculation Methodology
If a number n is composite, it must have at least one non-trivial factor less than or equal to the square root of n (√n). Testing up to √n via 6k ± 1 provides a deterministic proof.
Worked Example
Testing 97
Testing composite 91
Frequently Asked Questions
Why is the number 1 not considered a prime number?↓
The Fundamental Theorem of Arithmetic requires unique prime factorization. If 1 were prime, factorizations like 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3 would lose uniqueness.
Is 2 the only even prime number?↓
Yes. Every even integer greater than 2 is divisible by 2, making 2 the only even prime number.
Why do we only need to test divisors up to the square root of n?↓
Factors always occur in reciprocal pairs (a × b = n). If both factors were strictly greater than √n, their product would exceed n, which is impossible.
What is a limitation of trial division?↓
For cryptographic numbers with hundreds of digits, trial division is computationally intractable, requiring probabilistic tests like Miller-Rabin.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.