This post got me thinking: Is there a formula for finding a specific sequence of numbers in pi? How would I go about finding, say, “236592” without just calculating the the value of pi until I reach that specific sequence?
I don’t know of any formula but the easiest thing to do is go to this site (pi to 4 million digits) or another like it and do a “find.”
For what it’s worth, I didn’t find “236592” in there, but I’m assuming that was just a number you chose arbitrarily.
Would the base 16 BBP Formula suffice?
IANAM, but I thought that while there were formulas like BBP that allow you to start at any arbitrary spot and calculate what appears there, that meant only that you would discover what set of digits followed, not reveal a specified set.
I don’t think there is any way to find a specified set that is not embedded in a known expansion. In fact, wouldn’t that violate most of what we think we know about pi?
Nothing to contribute to the OP’s main issue, but I find it marvelous that you can just keep digging into the digits of Pi in the same way you can continue to “zoom in” on the sections of the Mandelbrot Set.
I’m not sure what distinction you’re making here. If you can start at an arbitrary spot and calculate what digits follow, can’t you just make your arbitrary spot the start of your specified set and then stop working when you’ve fixed the digits to the end of your specified set?
Anyway, to tell the truth, the only knowledge I have of the BBP algorithm (before this morning) is “having heard of it”. Wikipedia does have some info, however, on how to do digit extraction using it. Another article gives a clearer example of the technique, but using a different formula.
I don’t understand what you’re saying.
You can start at the 4 billion trillionth number. But you don’t know ahead of time what that number is until you calculate it. And even when you do you don’t know the next number until you calculate it. How does that help you find 236592?
What would it violate? We know that any specific sequence of digits exists in pi. We know that we can find it, albeit not quickly.
Only if I can get a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
I’m betting his (Exapno Mapcase’s) thinking goes like this:
IF there was a formula or algorithm where we could feed in an arbitrary digit sequence and get back the location of that sequence within the decimal expansion of pi WITHOUT having to perform the decimal expansion starting from “3.” & continuing until we encountered the sequence of interest …
THEN that would imply there’s some manner of predictable internal structure to the decimal expansion of pi. Which in turn would invalidate some existing knowledge about its random, non-repeating, normal, [insert fancy math term I don’t know] , etc., attributes.
His thinking seems reasonable on its face. And in all likelihood, the actual attributes of pi are such that an algorithm like that cannot every be constructed.
Ah. I was stubbornly misreading the OP as “know where you want to look, how to find what’s there without doing a full expansion” instead of “know what you want to find, how to know where to look without doing a full expansion”. On the latter, I got nothing.
It would violate PI being irrational.
For any specific pattern, you could set up an irrational number so that a particular pattern would be found at a specific spot. Like, if you wanted to find 8675309 at the twentieth (through 26th) digits of PI, you could add or subtract a rational number to PI to accomplish it - an irrational number plus a rational number is irrational.
But for a general formula to find an arbitrary sequence of numbers in an irrational decimal representation without calculating it implies a pattern that further implies rationality.
I have an excellent proof on my shirt slee- aw, crap. The maid just washed it. I’ll be back later.
No, we don’t. It’s widely believed, but it’s never been proven.
This I agree with; any string of digits that happens to be in there can be found by just enumerating digits till you find it.
But of course that’s precisely the sort of thing the OP isn’t interested in. They’re looking for a way to find the location of a desired string of digits more efficiently than by brute-force search.
This isn’t true.
Consider the number 0.01234567890001020304050607080910111213141516171819202122…
Give me any string of digits, and I can easily tell you a location where it will show up in this number. There’s a very simple formula that’ll do this for you.
But the number is irrational all the same.
The one and only answer is: Nobody knows any better way of doing it.
A quick Google for “search pi” brought up http://www.angio.net/pi/piquery which lets you search in the first 200 million digits of pi. The string 236592 occurs at position 140,928.
Oh, I know. It just seems reasonable to ask the question that if we have a slow method, does a fast method exist?
Walton Firm, that’s the brute force method. First compile digits of pi, then search for the sequence. Well, the brute force method would be compile the digits until you get to the sequence you want, but that’s close enough.
Is there a data compression subquestion here? If so, information theory prevents it being more economical (in general) to store the position of a sequence than just to store the sequence itself.
Very interesting. The string “11111111” occurs at position 159,090,113. Can I assume that 9 "1"s are the max?
8675309 occurs at 9,202,591… Hmmm, curious! Tommy Tutone were obviously advanced mathematicians.