What’s the longest repeating string of numerals found (so far) in the decimal representation of Pi?
We all know (or have been told) the numerals used to represent Pi (in decimal) never repeat. But there must surely be “strings” of numerals that are found more than once in the decimal representation of Pi.
For example, Pi = 3.14…54…54… or Pi = 3.14…4930495…4930495…
Does anyone know what this longest string of numerals may be?
I would think that by definition, because there are always more numbers in new patterns as you continue the expansion, it would be impossible to solve in the general case.
But I’m slightly intrigued. Might go see if there’s anywhere I can download ‘a million digits of pi’ or something and write a few quick programs to see how long a repetition I can track.
In the decimal expansion of pi, there must be repetitions of any length you specify. However I don’t know what the record would be for the longest one that’s actually been found.
Well, 756130190263 appears twice in the first million digits…
Here’s a pi search engine:
That’s true if pi is normal, which I don’t believe it’s actually known to be.
No, normalcy implies that every sequence appears with the same asymptotic frequency. But because there exist only a finite number (10[sup]n[/sup]) of distinct sequences of n digits, eventually at least one of these sequences must repeat.
Tucker’s link leads to the University of Tokyo (FTP), where you can get at least the first 200 million digits.
Right, so if there is a finite sequence of digits that never appears in pi, or even appears just a finite number of times, then pi would not be normal. (There are also other ways of violating normality; I was focussing on just the one.)
Pi is not yet known to be normal, though from all the digits calculated so far, it would be astounding if it weren’t.
How is that known to be true?
Agreed.
Because in a sequence of 10[sup]n[/sup]+n digits, there are 10[sup]n[/sup]+1 subsequences of n consecutive digits; clearly (since there are only 10[sup]n[/sup] distinct n-digit sequences) at least two of these are equal. As I understand it, this is what the OP is asking for: e.g., the sequence 756130190263, which occurs at digits 447673 and 857982.
Because there are 10[sup]n[/sup] possible sequences of length n, and there are more than 10[sup]n[/sup] such sequences in pi. If you want to find a sequence of n digits that is repeated, look at these sets of digits in pi:
1…n
n+1…2n
2n+1…3n
…
10[sup]n[/sup]n+1…10[sup]n[/sup]n+n
There are 10[sup]n[/sup]+1 of these, so there must be a repitition in there somewhere.
Ah, all right. I parsed borschevsky’s post wrong. I took him to mean, “for any sequence you find in pi, you’ll always be able to find it repeated somewhere”. But that’s not what he was saying; he was saying what you’re saying.
Never mind.
Why?
Is there any reason to believe that the properties of the decimal representation of pi are similar to the properties of a finite prefix?
Very well, let me amend that to: It would be astounding to me if pi were discovered not to be normal — especially in an arbitrary base like ten, which was an entirely human choice.
“You want a piece of me?” -?
“You want a piece of me?” -pi
I remember seeing (in the “pre-computing search” days), 10,000 digits of pi.
Somewhere in those first 10,000 digits, the number ‘9’ appears six times in a row. (Great place to round it don’t you think?)
Okay, it occurs pretty early in the pi sequence. From digits 762 through 767 you will find the ‘999999’ sequence.
Eric W. Weisstein. “Normal Number.”
What’s the likelihood that pi would be normal in base 10 but not absolutely normal?
And what is 2-normal?
In his 1988 paper on calculating the first 29 million or so places (reprinted in the Borwein’s Pi: A Source Book), David Bailey gives a formula for the expected number M of repeats of length n in d digits. This is M = 10[sup]-n[/sup] d[sup]2[/sup]/2. (Which should be easy enough to prove, though I haven’t bothered checking.) There’s also a simple formula for the associated variance: V = 11 M/9. Neither result is quite exact, but they are good approximations.
The longest repeated sequence Bailey found involved 15 digits, though he doesn’t say what this was or where it was found. That’s about what you’d have expected based on the formulae above.
Since then, the calculation of the expansion has been pushed out to 1.2 trillion digits. You’d expect to be finding something like a 24 digit repeat somewhere in that.