Prime Number Generator

This generator lists prime numbers — whole numbers greater than 1 whose only divisors are 1 and themselves. Ask for every prime in a range, such as 1 to 100 or the last million numbers below one trillion, or for the first N primes. You get the count, the smallest and largest prime found, their sum, and a comma-separated list ready to copy.

How to use this tool

1. Choose a range, or "the first N primes".
2. For a range, enter the start and end; both ends are included.
3. For the first N primes, enter N.
4. Copy the list from the results box, or read it in rows of ten in the table.

Formula

Sieve of Eratosthenes: for each prime p ≤ √(end), cross out p², p² + p, p² + 2p, … — every number left is prime.

Only primes up to the square root of the end are needed, because every composite number n has a factor no larger than √n. For ranges that start high, the generator runs a segmented sieve: it sieves only the numbers you asked about, using those small primes, so it avoids allocating every integer below a high starting point; preparing the base primes still adds work. For the first N primes it sieves up to N(ln N + ln ln N), a bound that is proven to exceed the Nth prime for N ≥ 6.

Example

From 1 to 100 there are 25 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. The smallest is 2, the largest is 97, and together they add up to 1,060.