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#26 2020-07-12 10:04:46

zetafunc
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Registered: 2014-05-21
Posts: 2,261
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Re: All primes formula

If
denote the prime numbers, then there exists a constant

and a sequence

such that
is the nth prime. It can be shown that this recurrence relation generates all the prime numbers -- however, the complexity of this problem (as is often the case with prime-generating constants) is determining the value of
to a sufficiently high degree of accuracy. The proof of this result uses Bertrand's postulate.

Here, I've used
to denote the floor of x (you can think of that as 'rounding down x to the nearest whole number') and
to denote the fractional part of x. So in other words, we'd have
and
.

The exact value of
can be represented as an infinite sum:

You need about 25 terms in the series above to get all the primes less than 100, for example.

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