Loop Sequences within Pi (FOUND ON THE WEB):

Dan Sikorski pointed out an interesting loop sequence within the decimal expansion of Pi.
If you search for 169, it appears at position 40. If you then search for 40, it appears at position 70. Search for 70, ... and so on.
The sequence Dan found is: 40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, 40...

My remark: the position is counted from the decimal point e.g. the string "1592" is located at position 3.

Prove: Not all number strings will loop.

I think this is probably the easiest way of showing it.

The digits of pi following the decimal point start .14159...

1 loops back to 1, which is fair enough.

But the same is also true of the numbers 14, 141, 1415, 14159 etc. 

So if any loop sends the sequence to any one of these numbers, then it gets stuck at 1 and can't loop back to the first number, as was to be proved.

Example: the strings 93, 932, 9323, etc. all start at position 14, so we search for 14 and at once get stuck at 1.

Edited on April 16, 2012, 12:54 am

Posted by broll on 2012-04-16 00:49:59