12
$\begingroup$

A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in integer coefficients) that allows computing the $n$th digit of $\alpha$ in the basis $b$ directly and without any need to computing the preceding $n-1$ digits of $\alpha$. It is also connected to the problem of normality of irrational numbers.

1- For which irrational numbers do we have a known Bailey–Borwein–Plouffe formula? For which of them do we have such a formula in the base $10$? (Please add a reference to a list if there is any).

2- Is there any known BBP formula for $e$ in an integer base $b$?

3- For which irrational numbers, the normality of the number in a given base is proved using a BBP formula?

$\endgroup$
4
$\begingroup$

Concerning (1), there is a large compendium of BBP formulas in this paper by Bailey. Regarding (2), it is strongly suspected (but not proved) that there is no BBP formula for $e$. Here is a quote from the paper by Bailey.

Powers of $e$ are specified here because, as far as anyone can tell (although this has not been rigorously proven), $e$ is not a polylogarithmic constant in the sense of this paper, and thus it and its powers are not expected to satisfy BBP-type linear relations (this assumption is confirmed by extensive experience using the author’s PSLQ programs).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.