math / Structures / free

Number Theory

Explore the patterns hidden inside whole numbers and primes.

Follow the mechanism1 / 3

Whole numbers can factor. A composite number is built from smaller factors, while a prime number cannot be split into anything but 1 and itself.

prime sievelimit 36 · step 1 / 3

Focus the idea

Cross the multiples of the active prime.Click a number to inspect its factors.

primecrossed multipleactive prime

primes found11

active prime2

multiples crossed17

selected30

Largest number in sieve 36Sieve checkpoint 2

Crossing out multiples of 2 removes composites. Numbers left after every smaller prime has had its turn are prime.

A composite number has a factor at most as large as √n, so primality only needs divisibility checks through that boundary. The sieve makes the same shortcut visible across a whole range.