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 Post subject: New integer sequence?
PostPosted: Wed Aug 03, 2011 10:25 am 
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I mentioned pi yesterday here. I must have dreamed about it or something as I woke up this morning with an idea for a potentially interesting integer sequence.

Let's look at the digits of pi. 3.141...

How many digits must you go into pi before the digit 3 is repeated? 10
How many digits must you go into pi before the digits 31 are repeated in sequence? 138
How many digits must you go into pi before the digits 314 are repeated in sequence? 2121
How many digits must you go into pi before the digits 3141 are repeated in sequence? 3497

See attached.
Attachment:
pi.png
pi.png [ 76.23 KiB | Viewed 1092 times ]


Ok... I guess its not new. Someone thought of this before me. I see this sequence listed here:
http://oeis.org/search?q=10+138+2121+3497

So what I'm curious about is as this sequence continues is there any particular function it should approach? Say an exponential function perhaps?

If there is such a function, would it be the same function found if we tried to do the same thing with the digits of e or sqrt(2)?

An example of the odd things that pop into my head.

Carl

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 Post subject: Re: New integer sequence?
PostPosted: Wed Aug 03, 2011 10:55 am 
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wwwmwww wrote:
So what I'm curious about is as this sequence continues is there any particular function it should approach? Say an exponential function perhaps?

If there is such a function, would it be the same function found if we tried to do the same thing with the digits of e or sqrt(2)?

Hi Carl, I think this touches a bit on information theory, Ramsey Theory and combinatorics.

I'm pretty sure this sequence should be approximately linear when plotted log(base 10). For any stream of digits that are "random" such as pi the probability to finding a sequence of N digits shouldn't be particularly significant until you have roughly 10^N digits.

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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 12:23 am 
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bmenrigh wrote:
wwwmwww wrote:
So what I'm curious about is as this sequence continues is there any particular function it should approach? Say an exponential function perhaps?

If there is such a function, would it be the same function found if we tried to do the same thing with the digits of e or sqrt(2)?
Hi Carl, I think this touches a bit on information theory, Ramsey Theory and combinatorics.

I'm pretty sure this sequence should be approximately linear when plotted log(base 10). For any stream of digits that are "random" such as pi the probability to finding a sequence of N digits shouldn't be particularly significant until you have roughly 10^N digits.
That assumes that pi is a normal number. So far, it hasn't been proven to be true for any base. For that matter, no irrational algebraic numbers or famous constants have been proven or disproven to normal numbers.

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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 8:37 am 
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GuiltyBystander wrote:
So far, it hasn't been proven to be true for any base. For that matter, no irrational algebraic numbers or famous constants have been proven or disproven to normal numbers.
Yes, I knew there was a term for this! I kept looking up things related to "transcendental number" to find the right term for this but I couldn't find one. At first I was going to describe it as a "cryptographically secure number" but then I realized that's probably not a good description either.

However, if what I described were to fail for Pi it would prove it is not a normal number.

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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 9:00 am 
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But doesn't all this depend on the fact that we just happen to express pi in base 10? What would happen if we expressed pi in base 2 (with "binimals" instead of decimals), or some other base?

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Last edited by KelvinS on Thu Aug 04, 2011 9:27 am, edited 1 time in total.

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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 9:25 am 
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How cool would it be if when listening to pi in base 2 you heard a voice or something? :lol:

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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 9:30 am 
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DudeHuLubeDaRubeCube wrote:
How cool would it be if when listening to pi in base 2 you heard a voice...
...saying, "Could somebody please make me a cup of tea."

Well they do say God works in mysterious ways! :lol:

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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 9:58 am 
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 Post subject: Re: New integer sequence?
PostPosted: Thu Aug 04, 2011 4:00 pm 
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bmenrigh wrote:
GuiltyBystander wrote:
So far, it hasn't been proven to be true for any base. For that matter, no irrational algebraic numbers or famous constants have been proven or disproven to normal numbers.
Yes, I knew there was a term for this! I kept looking up things related to "transcendental number" to find the right term for this but I couldn't find one.
lol, I too thought that "transcendental" was the term for this. I was getting quite annoyed that I couldn't find anything in the description about digit statistics.
Kelvin Stott wrote:
But doesn't all this depend on the fact that we just happen to express pi in base 10? What would happen if we expressed pi in base 2 (with "binimals" instead of decimals), or some other base?
Yes, but so far, pi hasn't been shown to be normal in any base. So far it looks normal based on first 30 million digits. Though I suspect that unless pi is specially constructed from the base you're in, it should be normal in all bases right?
DudeHuLubeDaRubeCube wrote:
How cool would it be if when listening to pi in base 2 you heard a voice or something? :lol:
Well, if it is truly a normal number, every conceivable (and probably the ones you can't conceive :P) sequence of bits and data will occur at some point in the infinite sequence. In fact, it will occur infinitely often. It's like an infinite number of monkeys typing on and infinite number of typewriters for an infinite period of time. Someone should convert pi to base 27 and see when Shakespear's works start appear :D

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