To find every prime below 900 you only need to cross out multiples of the numbers up to 30. The reason is short: any composite n ≤ 900 must have a prime factor ≤ √n ≤ 30, so it has already been struck out — that is the whole trick of the Sieve of Eratosthenes. The survivors thin out at exactly the rate the Prime Number Theorem describes, roughly one prime per ln x numbers around x, yet Euclid proved they never run out. Lay the integers along a square spiral and the primes gather, unbidden, along diagonal lines.
To find every prime below 900 you only need to cross out multiples of the numbers up to 30. The reason is short: any composite n ≤ 900 must have a prime factor ≤ √n ≤ 30, so it has already been struck out — that is the whole trick of the Sieve of Eratosthenes. The survivors thin out at exactly the rate the Prime Number Theorem describes, roughly one prime per ln x numbers around x, yet Euclid proved they never run out. Lay the integers along a square spiral and the primes gather, unbidden, along diagonal lines.
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