Something like this can be used for "proof of work" in the blockchain world.
A formula for the nth digit of π and π^n
91β100 of 143 posts
Re: A formula for the nth digit of π and π^n
#92Earlier quoted context omitted.
Not sure where you're seeing that. Did you misread a formula?
The very first denominator seen is (2Ο)^2n, the last denominator on that line is the same, the first denominator of the second line is Ο^2n+1 and the second denominator in the second line is the same. Are you not seeing that?
Are you talking about the inequality on the first page? That is at the end of a sentence that starts "The calculation is made from the two inequalities ...". The next sentence is "By isolating Ο in both cases, we can derive an approximation of the latter."
That is not the formula.
Re: A formula for the nth digit of π and π^n
#93Earlier quoted context omitted.
The article title says "powers of π", so presumably an arbitrary digit.
Iβm not sure thatβs right. The abstract and text say the nth digit of \pi^n, and the worked example on page 2 uses the same n=1000 to find the thousandth digit of \pi^{1000}.
Yes, it looks like you're right. But I don't understand the reasoning - is he actually saying you can use this formula to compute pi^n to n decimal places? So it doesn't produce arbitrary digits on their own, but excellent approximations to pi^n?
And this was by taking 4 terms of the zeta function infinite product expansion. 2xn digits would presumably be given by taking more terms (but quite a few, one for every prime under 100, so in some sense this method has bad convergence when you want the mth digit of pi^n and m >> n).
Re: A formula for the nth digit of π and π^n
#94Earlier quoted context omitted.
Why is zero different than any other number?
It acts as a terminator or terminal, see https://news.ycombinator.com/item?id=34391483 .
If 0 would act as a terminator, then 1.203 would be equal to 1.2.
The thing that is actually special about the digit 0 is that it is implied for all positions for which no digit is given. That is, when we write
1.2
β¦00000001.20000000β¦Re: A formula for the nth digit of π and π^n
#95The paper seems absolutely brilliant, but the grammar is very strange (thereβs even what appears to be a typo in the paper where he says βrand nβ instead of βrank nβ). Odd that he wouldnβt have worked with somebody with better written English before publishing.
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.
Can you insist the writing be improved for publication?
Re: A formula for the nth digit of π and π^n
#96Earlier quoted context omitted.
Iβm not sure thatβs right. The abstract and text say the nth digit of \pi^n, and the worked example on page 2 uses the same n=1000 to find the thousandth digit of \pi^{1000}.
Yes, it looks like you're right. But I don't understand the reasoning - is he actually saying you can use this formula to compute pi^n to n decimal places? So it doesn't produce arbitrary digits on their own, but excellent approximations to pi^n? And this was by taking 4 terms of the zeta function infinite product expansion. 2xn digits would presumably be given by taking more terms (but quite a few, one for every priβ¦
Yeah, my sense is that the author knows this is more of a neat trick than a "practical" way of calculating pi.
There's a bit on page 5 about how the Chudnovsky method totally blows this approach out of the water.
Re: A formula for the nth digit of π and π^n
#97This is a type of https://en.wikipedia.org/wiki/Spigot_function if you want to lookup more info.
Re: A formula for the nth digit of π and π^n
#98Earlier 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¦
The goal of publishing is that others can read your work. If it's badly written it will be hard to understand, and extra hard for non-natives.
Re: A formula for the nth digit of π and π^n
#99Plouffe ... 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.
Re: A formula for the nth digit of π and π^n
#100Earlier quoted context omitted.
But why?
For me, it stimulates a mental shift that could be good exercise elsewhere: instead of storing the string, you find the string somewhere and store its index. It also stimulate the imagination: what other transcendental numbers might this work with? How long do you have to search in the digits to find your string? What can you say about the size of the index (how far you searched) vs your string length? Etc. It's pattβ¦
I don't want to be a buzzkill, but to me this seems uninteresting, you need more information to store the index than the data, end of story.