A formula for the $n^{\rm th}$ decimal digit or binary of $π$ and powers of $π$ — HN submission

Sun, 15 Jan 2023 13:12:38 GMT

A formula for the nth digit of 𝜋 and 𝜋^n

268 points; 133 discussion replies on HN. Paper and source history. A submission is not an endorsement.

A formula for the $n^{\rm th}$ decimal digit or binary of $π$ and powers of $π$ — mentioned by schoen

Sat, 28 May 2022 01:51:38 GMT

I was very surprised to learn that Plouffe has actually also devised a formula that gives the nth decimal digit! It looks like this was a much more recent discovery. https://mathworld.wolfram.com/Digit-ExtractionAlgorithm.html https://arxiv.org/abs/2201.12601 I first wrote that I thought this was asymptotically much faster than computing the intermediate digits, but somewhat slow because it requires computing Bernoulli numbers as one input. But in Plouffe's PDF, he says that actually the convergence is very slow and you would do significantly better computing the intermediate digits with…

Paper and discussion history. A mention is not an endorsement.

A formula for the $n^{\rm th}$ decimal digit or binary of $π$ and powers of $π$ · HackerPapers discussions

A formula for the $n^{\rm th}$ decimal digit or binary of $π$ and powers of $π$ — HN submission

A formula for the $n^{\rm th}$ decimal digit or binary of $π$ and powers of $π$ — mentioned by schoen