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$\frac{\pi}{3}e$ is approximately equal to:

2.84657807422452235515451695651552483167829617858837165395986704339307620371911026919085462323682464797125831417055915588210706253023200799687278781793023471514007199829654447617582233348895922031024237096797835826800351909013449542434666218846881388726409761904569405458137658722823724181557439473446917219291006835569333157358766322654791237966966946623595073544349432678461546724384333992745425590086011913751990685057546632872448319697537470763226921527595140181054405265139836753917775379775697089746424785856104245156874061419780994637730604531480206814256732829792410814871969646790020050463395548624913283274551800217089819065169939172989203287463740049502718515202660522891769528913026965548612122163723732564317175312804182112811958680271600871936194429224533157202322236063849980512815661793560779711542728888881905074320...

While $e$ contains a lot of repeating digits - I'd normally expect this to disappear once $\pi$ is introduced.

It almost looks like the further along you go, the greater the degree of repetition.

Is this just due to an interesting interaction that occurs only with decimal approximations of pi and e, or is there more going on?

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closed as not a real question by BenjaLim, AD., Thomas, Arkamis, rschwieb Oct 24 '12 at 15:31

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4  
It does seem curious to me. I didn't see a lot of repeated digits, especially so "early on", for $\pi$ or for $e$. –  coffeemath Oct 24 '12 at 12:30
3  
Any statistics on the frequency of 2-string, 3-string, ..., n-string? (substrings made of $n$ identical numbers) –  FrenzY DT. Oct 24 '12 at 12:34
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I believe this is more like numerology than mathematics. –  AD. Oct 24 '12 at 12:35
1  
I'll code that up this evening if someone doesn't beat me to it. –  PhonicUK Oct 24 '12 at 12:36
4  
This question seems to be another example of how people greatly underestimate just how "clumpy" uniformly random data is. –  Hurkyl Oct 24 '12 at 16:26

1 Answer 1

Considering the fractional part shown, there are $77$ occurrences of repeated digits (a digit followed by the same digit) in a string of $830$ digits. The expected number of repeated digits would be $82.9$ with a $\sigma$ of $8.64$. This is $0.683\sigma$ below the mean.

Correction

The search I performed did not count all repeated digits, as I had thought. It started the next search after the end of the previous search, so it missed the second pair that occurs in a triple. The count of doubles is actually $86$. This is $0.359\sigma$ above the mean. Not terribly significant, but above the expected number.

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3  
I guess we should change it to "Why does $\frac{\pi}{3}e$ contain so little repeating digits?" –  EuYu Oct 24 '12 at 15:35
1  
@EuYu: sorry, my count was off because the search did not behave as I expected it to. –  robjohn Oct 24 '12 at 16:18
    
For future reference, the proper Perl-style search string to find repeated digits is (.)(?=\1) This finds only the first of the repeated digits, so that the subsequent searches can find the next pair, even in triples, etc. –  robjohn Oct 24 '12 at 16:48

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