Prime Sieve Explorer

Intro to Proofs · Axiom Academy

Discover the ancient algorithm that reveals the primes — and the mysterious way they thin out. Finding primes by crossing out everything that isn't one A prime is a whole number greater than 1 whose only divisors are 1 and itself — 2, 3, 5, 7, 11, and on. Over 2,000 years ago Eratosthenes found a beautifully simple way to sift them out of a whole grid of numbers at once, with no division and no guessing. You don't have to take it on faith: watch it run, then go explore what the survivors look like. Watch the grid of 1 to 60 . Keep the first uncrossed number — it's prime — then cross out all of its multiples. Do that for 2 , then 3 , then 5 , then 7 , and every number left standing is prime. (Why stop at 7 ? Any composite up to 60 has a factor , so once the multiples of 2,3,5,7 are gone, nothing composite can survive.) Crossing out the multiples is the whole algorithm — the primes are simply whatever the sieve never strikes. How are the primes spread out? Now that the sieve hands us every prime, look at how they're distributed. Slide the cap to count the primes in each block of ten. The bars lean shorter as you climb — primes get rarer the further out you go — yet a new one always turns up. There's no formula that tells you where. 2 is the only even prime — every other even number is a multiple of 2 . Primes grow scarcer as the numbers get larger. No simple rule predicts where the next one lands. The gaps between consecutive primes keep changing.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.