The ones that come up in class, at 2 a.m., and everywhere in between.
Q·01What is a prime number?
A whole number greater than 1 whose only positive divisors are exactly two: 1 and itself. 2, 3, 5, 7, 11, 13… 2 is the only even prime, since every other even number has 2 as a third divisor. Numbers with more than two divisors are composite; 1 has only one and is neither.
Q·02How do I check if a number is prime by hand?
Shortcuts first: even → composite; digit sum a multiple of 3 → composite; ends in 0 or 5 → composite. Then trial-divide by 7, 11, 13… only up to √n. If nothing divides, it's prime. For 89: √89 ≈ 9.4, so you only test 2, 3, 5 and 7 — none divides, so 89 is prime. That's exactly the run this tool writes out for you.
Q·03Is 1 a prime number?
No. Primality demands exactly two divisors, and 1 has just one — itself. It's called the unit. Some old texts listed 1 as prime, but the modern convention protects unique factorization: 6 must equal 2 × 3 in exactly one way, which would break if 1 counted.
Q·04Why only test divisors up to the square root?
If n = a × b with a ≤ b, then a × a ≤ a × b = n, so a ≤ √n. Every composite has a factor at or below its square root; if no candidate ≤ √n divides, no larger one can either. This collapses a 10-digit number from a billion tests to about 31,623 — and the 6k ± 1 wheel removes two-thirds of those.
Q·05How do I find all primes between two numbers?
The Sieve of Eratosthenes: list the integers, strike the multiples of each prime up to √(upper bound), and whatever survives is prime. This tool uses the segmented version — it only sieves base primes up to √(upper) and screens your window directly, so primes near 1012 cost no more than primes near 100.
Q·06What is prime factorization?
Every integer n > 1 is a product of primes in exactly one way, up to order — the fundamental theorem of arithmetic. 360 = 2³ × 3² × 5, and no other prime list multiplies to 360. From it you can read the divisor count τ, the divisor sum σ, and Euler's totient φ — the FACTOR tab computes all three.
Q·07Are there infinitely many primes?
Yes — Euclid proved it around 300 BC: assume a complete list, multiply every prime on it and add 1; the result is divisible by none of them, so either it's prime or it has a new prime factor. Contradiction. The primes also thin out predictably: π(n) ≈ n / ln n.
Q·08What are twin primes?
Prime pairs separated by exactly 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31)… Whether infinitely many exist is the twin prime conjecture — open since 1846 — though it's now proven that prime gaps of at most 246 occur infinitely often. The RANGE tab counts the twin pairs in your window.
Q·09What is the largest known prime?
282,589,933 − 1, a Mersenne prime with 24,862,048 digits, found by the GIMPS project in December 2018. It's only the largest known, never the largest — Euclid forbids a largest. Mersenne primes dominate the record books because the Lucas–Lehmer test checks them so efficiently.
Q·10Is the checker free — does my number leave the page?
Free, no sign-up, no locked features. Every division runs locally in your browser with integer-exact arithmetic, nothing is recorded or transmitted, and the page keeps working offline once it has loaded.