Prime Number Calculator — primality test, factorization, range & N-th prime

Real-world use cases

• Pick safe primes for an RSA homework key

  A cryptography assignment asks you to generate an RSA key by hand with two distinct primes p and q. You guess two candidates around 6-7 digits, paste each into the primality-test mode, and get an instant yes/no — no "probably prime" hedging, because Miller–Rabin is deterministic at this size. If one is composite the tool shows the smallest factor so you know to bump it up. Then you compute n = p·q, and if you ever need to double-check the modulus did not accidentally share a factor, the factorization mode confirms n cleanly splits back into exactly p and q.

• Check a student's prime factorization answer

  A worksheet asks "write 1260 as a product of primes". A student writes 2^2 · 3^2 · 5 · 7. You paste 1260 into the factorize mode and read back 2^2 · 3^2 · 5 · 7 — match. When a student gets it wrong (say they wrote 2^3 · 3 · 5 · 7), the side-by-side comparison makes the error obvious: the exponent on 2 is wrong. Because the output is in the same exponent notation textbooks use, you are grading against the exact form you taught, not a flat list of repeated factors.

• Generate a prime test set for a coding interview

  You are writing a `isPrime()` function for a coding-interview prep session and need correct expected values. Use the range mode on [2, 100] to get the canonical first 25 primes for a unit-test fixture, then grab a few large primes from the N-th prime mode (e.g. the 10000th prime is 104729) to test the path your trial-division loop is slowest on. Having a trusted oracle means your test failures point at your bug, not at a wrong expected value you typed from memory.

• Find a prime modulus for a hash table

  You are sizing a hash table and the textbook advice is "use a prime larger than your expected load to reduce clustering". You expect about 30,000 entries, so you open the range mode on [30000, 30050] and pick the first prime it lists (30011). Or use the N-th prime mode to walk primes near a power of two. Either way you get a verified prime in one click instead of eyeballing a number and hoping it has no small factors.

• Sanity-check a number-theory conjecture by hand

  You are exploring twin primes (pairs that differ by 2) for a math club. Run the range mode on [1, 200] and scan the list for adjacent entries two apart — 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. Because the sieve returns the complete, exact list with no gaps or probabilistic skips, you can trust the pattern you spot is real and not an artifact of a fast-but-approximate test.

Common pitfalls

• Treating 1 as prime. It is not — a prime needs exactly two divisors and 1 has only one. Many hand-written sieves and homework answers wrongly include 1; this tool excludes it and tells you why.

• Confusing "is prime" with "has no small factors". A number can pass a quick check against 2, 3, 5, 7 and still be composite (e.g. 121 = 11^2). Use the primality mode, which tests properly, rather than eyeballing small divisors.

• Expecting the factorization of a giant prime to be instant. Proving a large number has no factors is the hard direction; if you only need yes/no, the primality-test mode is far faster than the factorize mode because it never has to find the factors.

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