20060201, 01:23  #1 
"Mike"
Aug 2002
2^{2}×7×281 Posts 
Shortest sequence of numbers not found in M43
We wrote a simple little program to search the decimal expansion of M43 and find the shortest sequence of digits that doesn't appear. The shortest appears to be "120423". Our questions are:
For a number that is 9,152,052 digits long we really expected at least all 6 digit numbers to be found. Is there any way of saying we'd expect a certain length of digits to appear, compared to maybe a random number of equal length? What relationship does the decimal expansion have to randomness in general? Say, up until December, if you gave someone the decimal expansion, would it have appeared to be a random stream of digits? Is there anything interesting about analyzing the distribution of digits in M43? Apologies in advance for errors in terminology, etc. 
20060201, 05:21  #2 
"William"
May 2003
New Haven
2^{3}×5×59 Posts 
We have 9,152,047 6 digit sequences.
If we fix our target sequence and pick a random starting point, we expect to match once in a million, and miss with probability 0.999999. The probability a fixed target sequence will miss every time is (0.999999)^9152047 = 0.000106 There are one million possible target sequences, each of which misses with probability 0.000106, so the expected number of missing sequences is 10^6 * 0.000106 = 106. So it isn't surprising that some sequences are missing  we should expect about 100 of them to be missing. If your definition of shortest means that every sequence beginning with "0" is found, that is surprising. I would expect about 10 of the 100 missing sequences to begin with a 0. William 
20060201, 07:15  #3  
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
Quote:
I don't think they're independent. 

20060201, 13:54  #4  
"William"
May 2003
New Haven
938_{16} Posts 
Quote:
But Expectation is a linear operator. Suppose the joint probability of A and B occuring or not is the matrix Code:
NotA Yes A Not B a b Yes B c d 

20060201, 15:40  #5  
"Mike"
Aug 2002
2^{2}×7×281 Posts 
Quote:
120423 131185 133219 162528 169112 171781 189328 202575 208588 209102 213227 214982 220868 253046 268050 287999 307795 316010 320114 332515 339446 381260 402272 414817 428888 429422 430029 430042 446894 454976 455286 482620 482932 485617 490494 506515 512136 523283 533259 536160 542009 555766 556540 566225 575486 585692 593704 598282 603914 607593 607916 610860 614841 634921 636073 638448 651783 660030 660498 667169 693473 699456 718760 720316 724821 725651 729367 734332 738205 753655 753934 757274 762406 771597 793060 800048 810588 814332 827045 830063 854597 869159 894255 904309 904615 915337 923155 933500 935084 973922 980845 988256 992864 997763 

20060201, 18:35  #6 
∂^{2}ω=0
Sep 2002
República de California
3×7×19×29 Posts 
Related questions:
 How many of each decimal digit are there in M43?  What are the lengths of the longest strings of the same decimal digit that occur? (e.g. longest string of 0s, 1s, etc)  What is the longest string of consecutive digits modulo 10? (i.e. substring of ...0123456789012345678901234567890123456789...) 
20060201, 19:02  #7  
"Mike"
Aug 2002
2^{2}·7·281 Posts 
Quote:
1: 914272 2: 916362 3: 913997 4: 914191 5: 916441 6: 915744 7: 915905 8: 916856 9: 914816 

20060201, 19:15  #8  
"Mike"
Aug 2002
2^{2}×7×281 Posts 
Quote:
1: 9 2: 7 3: 6 4: 7 5: 7 6: 6 7: 7 8: 8 9: 7 

20060201, 19:20  #9  
"Mike"
Aug 2002
2^{2}·7·281 Posts 
Quote:
1: 1234567 2: 2345678 3: 345678 4: 456789 5: 567890 6: 6789012 7: 789012 8: 890123 9: 9012345 

20060202, 06:54  #10 
Jun 2005
2×191 Posts 
What's the longest sequence that appears twice?
3 times? Drew 
20060202, 15:04  #11 
Jun 2005
Near Beetlegeuse
2^{2}·97 Posts 
Let me get this right. Wblipp (who experience tells us knows what he is talking about) says that he would expect about 100 6digit sequences to be missing from M43.
While Xyzzy (who experience tells us does know how to programme stuff) says that he can find less than eight dozen valid 6digit strings in the decimal expansion of M43. Surely one of these must be way off the mark ? 
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