An iteration produces all the prime numbers

[Valentin Albillo]

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Hi, EdS2:

(02-13-2021 06:05 PM)EdS2 Wrote:  This very much appeals to me: initially it's rather a surprise.

Take the number 2.920050977316134712092562917112019... and perform the following iteration:
x := IP(x)*FP(x)+IP(x)

(Where IP is the integer part and FP the fractional part.)

There are infinitely many such constants that produce primes (this one, Mills' constant, etc) but though they're somewhat interesting from a 'recreational maths' point of view, they're useless as prime-producing constants because you need to know in advance the prime sequence to compute them, which makes the subject rather circular:

      You can use the constant to produce primes but you need the primes to produce the constant.

It would be quite another matter if any such constant could be computed to arbitrary precision some other way without involving the primes, say as a limit, an infinite summatory, an integral ... Alas, that's never the case so far and it's highly dubious it will ever be.

Regards.
V.