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A formula for the nth digit of π and π^n
131β140 of 143 posts
Re: A formula for the nth digit of π and π^n
#132Earlier quoted context omitted.
Not this one, but algorithms that can calculate the specific digit positions but nothing else are indeed used for world records, mainly for the verification. If two radically different algorithms converge into the same digits at something like the trillionth position then you will have a high confidence for the rest of digits.
Also wouldnt this be highly parallelizable?
Yes, that fact gave rise to projects like PiHex [1]; hexadecimal digits of pi at around 10^15-th positions were already known in 2000, while the current full-digit record is only for first 10^14 decimal digits.
Re: A formula for the nth digit of π and π^n
#133Earlier quoted context omitted.
You have no idea how atrocious the English is in papers that I see as a reviewer. And depending on journal I don't even get to reject it for that as long as the science is sound.
> And depending on journal I don't even get to reject it for that as long as the science is sound. and you shouldn't. As long as it is somewhat legible, forcing people who are not native English speakers to conform to another language 100 % in order for their science to be published is horrendous. There was a time 300 years ago, where great thinkers who did not speak French or German could not publish their thoughtsβ¦
> and you shouldn't
If you say so - but honestly, why not just publish it in French then (I think the author is French)? If I were publishing something in French - technical or scientific or otherwise - Iβd want it to be reviewed by a native speaker.
Re: A formula for the nth digit of π and π^n
#134Something like this can be used for "proof of work" in the blockchain world.
That's an interesting point. You could use this to hash values with infinitely scaling difficulty by providing an offset to n where n = difficulty + n. The only issues is collision since the output is always 0-9 but if your POW check is against multiple digits then it might work? Again you could still have potentially infinite repetitions of sequences of digits with different inputs.
Yes, I meant for multiple digits.
Basically the work will be give me starting index from which, next N digits are = "45334138023580". Finding the first digit can take ages while verification is O(1)ish.
Imagine that to provide storage systems similar to IPFS, but the blockchain network only stores the metadata and no data!
Re: A formula for the nth digit of π and π^n
#135I don't understand this claim on the bottom of page 1 (emphasis added):
> It is easy to verify that with n = 1000, the error on Ο is of the order of 0.293193 x 10^-303, which is less than 2^-1000. So the thousandth position of this expression is the 1000th bit of the number Ο.
How does that follow? Take the number 1 - 2^-n. The difference between this number and 1 is 2^-n by definition, so it can be made arbitrarily small by varying n, but all bits are wrong. Addition can propagate all the way to the "decimal" point (and even beyond), so an error bound doesn't normally say anything about individual bits or digits. What am I missing here?
Re: A formula for the nth digit of π and π^n
#136Is there a physical limit to how many digits of pi can ever be computed/represented in the universe? For example, letβs say we need one atom for each digit of pi that we want to store, the max limit of digits of pi would be something like the total number of atoms in the universe, minus the atoms required to compute and store the digits. Has that been studied/calculated?
This sounds like a lot of work, when the physical limit of how many decimal places you need in our universe given the Planck length is only 63 or so.
What do you mean that we βonly needβ 63 decimal places?
A computer can calculate and hold a lot more than 63 digits.
So I guess my question is more like: if the universe was just one big computer dedicated only to calculating (and storing) digits of pi, how many digits could it get to calculate, max?
Re: A formula for the nth digit of π and π^n
#137Plouffe ... He used to teach in my university. He is a fantastic man. This is serious, he is crazy about numbers. He is one of the guy behind OEIS (oeis.org). That is not the first formula he found about pi and some of the previous one had been used to break world record about the number of known decimals.
Probably this is a good time for your irregular reminder that Plouffe has something of a feud with Bailey and Borwein: https://groups.google.com/g/sci.math.symbolic/c/a3kVKVYJhgc?... In my very personal opinion, the overall publication history of the three provides some rather good evidence for Plouffe's claims.
Oh thank you for the link !
In fact, I wasn't sure about the full story but the fact that Wikipedia repeatedly told that Plouffe alone discover the formula makes me think that there is a story behind that.
Re: A formula for the nth digit of π and π^n
#138Earlier quoted context omitted.
That's an interesting point. You could use this to hash values with infinitely scaling difficulty by providing an offset to n where n = difficulty + n. The only issues is collision since the output is always 0-9 but if your POW check is against multiple digits then it might work? Again you could still have potentially infinite repetitions of sequences of digits with different inputs.
Yes, I meant for multiple digits. Basically the work will be give me starting index from which, next N digits are = "45334138023580". Finding the first digit can take ages while verification is O(1)ish. Imagine that to provide storage systems similar to IPFS, but the blockchain network only stores the metadata and no data! https://github.com/philipl/pifs
Cool little project, cheers for the link.
Definitely a neat concept, would be interested to see it applied like that.
Re: A formula for the nth digit of π and π^n
#139Earlier quoted context omitted.
This sounds like a lot of work, when the physical limit of how many decimal places you need in our universe given the Planck length is only 63 or so.
What do you mean that we βonly needβ 63 decimal places? A computer can calculate and hold a lot more than 63 digits. So I guess my question is more like: if the universe was just one big computer dedicated only to calculating (and storing) digits of pi, how many digits could it get to calculate, max?
Thereβs no possible circle in our universe that needs more than those digits of PI for accurate calculations. Obviously itβs fascinating to study Pi for theoretical reasons and Iβm sure someone can tell me there are also practical reasons to do so. I was just surprised to find out that, given the Planck length, the practical need for precision is astonishingly low compared to whatβs been calculated, even moreso compared to your theoretical upper limit!
Re: A formula for the nth digit of π and π^n
#140The paper begins by citing an equation that isn't explicitly in the book referenced. I wrote up some notes filling in a few missing details.
https://www.johndcook.com/blog/2023/01/16/pi-bernoulli-numbe...