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Prime numbers

A prime is a whole number bigger than 1 whose only factors are 1 and itself. The atoms of arithmetic.

Sieve of Eratosthenes

Cross out the multiples — what's left is prime.

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100

Currently crossing multiples of 1. Click "Cross out" to keep going. After multiples of 7 the rest are all prime — the square root of 100 is 10, and any composite up to 100 must have a factor ≤ 10.

Why are they interesting?

Every whole number bigger than 1 can be broken down into a product of primes — and there's only one way to do it (up to order). This is the Fundamental Theorem of Arithmetic. Primes are the building blocks of every other number.

The first few primes

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

Why isn't 1 prime?

By definition. If 1 were prime, every number would have many different prime factorisations (you could throw in any number of 1s). Saying 1 isn't prime keeps factorisations unique.

How do you check if a number is prime?

  1. Try dividing by 2. If it works and the number isn't 2, it isn't prime.
  2. Try 3, 5, 7, 11, … (only primes).
  3. You only have to check primes up to √n. If nothing divides cleanly, it's prime.

How many primes are there?

Infinitely many. Suppose there were only finitely many. Multiply them all together and add 1. The result isn't divisible by any of them (it leaves remainder 1) — so it has a prime factor not on your list. Contradiction. So the list never ends.

Quick check

  1. Is 51 prime?
  2. Write 60 as a product of primes.
  3. What's the smallest prime bigger than 100?

Answers: No (51 = 3 × 17), 60 = 2 × 2 × 3 × 5, and 101.