List Of Prime Numbers

What this calculator does

So, you want to see all the prime numbers up to a certain point, laid out in one list. This tool does that. Choose an upper limit and it lists every prime from 2 up to that limit.

You pick the limit from a dropdown, in steps up to 1000, and the tool fills in the primes. It is the list-at-a-glance companion to the single-number prime check.

How to use it

1. Choose the upper limit from the dropdown.
2. Press Calculate to see all the primes up to it.

The primes up to your limit

A prime number is a whole number greater than 1 divisible only by 1 and itself. The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19, and carries on without end. The first prime is 2, which is also the only even prime, since every other even number is divisible by 2. After that the primes are all odd, though of course not every odd number is prime. This tool walks through the numbers up to your chosen limit and collects the ones that are prime into a single list.

The classic way to list primes

There is a beautifully simple method for listing primes up to a limit, invented in ancient Greece by Eratosthenes and still called the sieve of Eratosthenes. You write out all the numbers up to your limit, then repeatedly cross out the multiples of each prime: first all the multiples of 2 beyond 2 itself, then the multiples of 3, then of 5, and so on. Whatever survives the crossing-out is prime. It is called a sieve because it strains the composite numbers away and leaves the primes behind. That is the natural way to generate a whole list of primes at once, rather than testing each number on its own.

How the primes thin out

One thing a list like this makes visible is that primes gradually become rarer as the numbers grow. Between 1 and 10 there are four primes, but between 90 and 100 there are only one. They never stop appearing, since there are infinitely many, but they spread further apart on average the higher you go. Seeing the primes up to 100, then up to 1000, side by side gives a feel for this slow thinning, one of the first patterns anyone notices about the primes and a starting point for a great deal of number theory.

A worked example

Choose a limit of 20. The tool returns 2, 3, 5, 7, 11, 13, 17, 19: the eight primes up to 20. Notice 2 is the only even one, and that numbers like 9 and 15 are left out, because 9 is 3 times 3 and 15 is 3 times 5, so neither is prime.

Questions people ask

What does this tool list?

Every prime number from 2 up to a limit you choose, in order.

Why is 2 the only even prime?

Because every other even number is divisible by 2, so it has a divisor besides 1 and itself and cannot be prime.

What is the sieve of Eratosthenes?

An ancient method that lists primes by crossing out the multiples of each prime in turn. Whatever is left uncrossed is prime.

Do primes ever run out?

No. There are infinitely many primes, though they grow rarer on average as numbers get larger.

How do I check just one number?

The is it a prime number calculator gives a yes or no verdict for a single number.

References

On listing primes. The sieve of Eratosthenes finds all primes up to a limit, and the primes continue without end.

1. "Sieve of Eratosthenes," Wikipedia, on the ancient algorithm for finding all prime numbers up to a given limit.
2. "Primes and Prime Factorisation," Australian Mathematical Sciences Institute, on prime numbers, the sieve, and Euclid's proof that they are infinite.