Execute high-precision primality checks, run cryptographic tests, compute factorization, and explore advanced number theory.
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Primality testing determines whether a given number is prime. Modern tests range from simple trial division to advanced probabilistic and deterministic algorithms suitable for cryptographic applications.
Prime numbers are the foundation of number theory and play a critical role in cryptography (RSA), random number generation, hashing algorithms, and many areas of computer science and mathematics.
Perform high-precision primality testing on any integer using multiple algorithms including Trial Division, Miller-Rabin, and Fermat tests. Instantly check if a number is prime, analyze its factors, and explore deep number-theoretic properties.
Ideal for mathematicians, competitive programmers, cryptography enthusiasts, and students. Supports single numbers, batch testing, prime gaps, nearby primes, and full diagnostic reports with educational explanations.
Choose from Simple, Probabilistic (Miller-Rabin), Deterministic, Cryptographic-grade, and Fermat primality tests with automatic algorithm selection.
Get prime factorization, smallest/largest factors, divisor count, sum of divisors, Euler’s Totient (φ), Möbius function (μ), and classification in one click.
Find previous and next primes, calculate prime gaps, and generate lists of nearby primes for deeper pattern analysis.
Test multiple numbers simultaneously and export results as JSON, CSV, TXT, or formatted HTML reports for research and documentation.
Testing extremely large numbers (especially over 10^15) can be computationally intensive. Probabilistic tests like Miller-Rabin are very fast but technically return “probably prime”, while deterministic tests guarantee accuracy within certain bounds.
Pro Tip: For numbers below 2^64, use deterministic Miller-Rabin with specific witness sets for guaranteed correctness. For cryptographic purposes, always combine multiple tests and consider certified primality libraries for very large numbers.
Batch testing of many large numbers may require significant processing time and memory. Enable optimization options when working with big integers.