Formula for primes

From Simple English Wikipedia, the free encyclopedia

Willan's Formula is a formula that can find the nth prime number.

Proof[change | change source]

Let's first start with the .

Wilson's theorem says if is divisible by , than is either a prime number or , meaning when is prime, is an integer.

It would be much easier if the formula gives a number instead of checking if the number is an integer, and we can do this with the part.

The reason the formula has multiplied by the part is because when is an integer, will give or .

When squaring the result then will equal when is an integer.

By flooring this, the only results are when is an integer and when it isn't, leaving.

The will add s for the primes - and and will sum up to the .

The in short will give if and when .

Take the of both sides where is the nth prime number:



gives the number , and the is because when reaches , the function doesn't add 1. The formula adds up to is because Bertrand's postulate says is bigger than the nth prime number.

And finally, is added because of the .[1]

References[change | change source]

  1. An Exact Formula for the Primes: Willans' Formula, retrieved 2022-11-01