Looping numbers in pi

  1. I noticed that there are some odd looping numbers in pi

    Following the rule that : the string becomes the position which becomes the string ( I'll use " Sn " for string number and " Pn " for position number, ( after the decimal ) becomes the next string to locate ( Sn → Pn → Sn )

    The process is easy, you start with the number and find it's position in pi digits, the position of the number becomes the next string to search for.

    For smaller numbers, ( excluding the self-locating digit 1 ) most of these series grow rapidly, however, certain numbers, like 16, 19, 23, 37, 40, 169, will loop on themselves following the rule of Sn → Pn → Sn

    So far I have found that

    19 loops on 37 ( 19*37 is a also a figurate triangular number )

    16 and 23 loop on 40

    169 loops on 40

    37 and 40 loop back to themselves
    --------
    37 → 46 → 19 → 37 →∞

    40 →70 →96 → 180 →3664 →24717 → 15492 → 84198 → 65489 →3725 → 16974 → 41702 → 3788 → 5757 → 1958 →14609 →62892 → 44745→ 9385 →169 →40→∞

    -----------

    So, does this mean that the above two series { 40 → ... 40 → } and { 37 → 37 } contains all looping numbers ?

    What is the reason for this phenomenon in pi digits ?

    How many numbers are there that will do this in pi ?

    Thanks :)
     
    Last edited: Feb 24, 2014
  2. jcsd
  3. So far I have done the first 50 numbers, this is some of what I have found so far

    16, 19, 23, 37, 39, 40, 43, 45, 46 will all end up in a loop

    Something neat:

    19*37 are factors of the triangular figurate number 703

    Number 19 will loop and have 46, 37, and 19 in the series of the loop

    19 further iterations of the rule will end up at number 37

    Number 37 loops on itself with 46, 19 and 37 in the series in the loop

    10 more steps is number 46, which loops on itself with 19, 37 and 46 in the loop

    ---------

    Numbers 16, 23, 39, 40, 43 all have the same loop as well, consisting of 20 numbers, in the series of the loop.

    1, 14, 21, 45 all loop back to the first self-referencing digit in pi, 1

    -----

    I'm just doing this out of boredom in case you're wondering
     
  4. A similar thing is here: http://oeis.org/A232013.

    Since the digits of pi are probably pseudorandom, I would expect the same sort of behavior from pi, on average, as I would in any random string of digits. But, off the top of my head, I don't know the probability that any particular number is involved in a loop, the expected length of a loop, or anything like that. I'm not even sure if it has been studied. It might be worth trying.
     
  5. Also some neat little facts about figurate numbers and repunits in pi, forgive me if this is known already

    Of the series { 111, 222, 333, ...., 777, 888, 999 }

    This starts with the string 111 at the 153rd digit, and ends with the string 999 at the 762nd position

    If you write the largest position numbers first, starting with the 4751st position locating the string 888, it will produce, from largest to smallest position value, the string of:

    888, 444, 666, 222, 333, 777, 999, 555, 111

    Which are split by even and odd numbers, evens first

    --------------

    As a rule, any 3 digit repunit can be divided by the sum of it's own digits to yield 37, example:

    666

    666 / ( 6 + 6 + 6 ) = 37

    The first two prime numbers are 2 and 3, and the positions of the strings 222 and 333 are 37 numbers apart

    222 at the 1735th digit
    333 at the 1698th digit

    1735 - 1698 = 37


    --------------------

    2701 is another figurate triangular number, which can be written as the sum of four other triangular numbers :

    666 + 666 + 666 + 703 = 2701

    If you locate the number 2701 in the digits of pi, you'll find that if you add the position numbers:

    2 is at the 165th position
    7 is at the 166th position
    0 is at the 167th position
    1 is at the 168th position

    165 + 166 + 167 + 168 = 666

    -------------

    222,111 is the 666th triangular number, the sum of the numbers 1 - 222,111

    The string 1112 is located at the 12701st digit in pi

    --------------

    Incidentally, many early civilizations had an interesting system of what could be considered an early form of encryption, usually referred to as " Isopsephy " and " Gematria " , for Greek and Hebrew, respectively, where each letter in the alphabet is assigned a number.

    I discovered that the total sums of the alphabets, both Greek and Hebrew, are identical to the the sum of the repdigits 111 through 999 , which is 4995

    Using figurate numbers is one way societies like the Greeks taught geometry

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    Feel free to move this if it's in the wrong section
     
  6. That's it, thank you.

    I was thinking about how Archimedes was trying to get an accurate value for pi with shapes, inscribing polygons and circles, and became curious how the numbers which produce shapes ( figurates ) might be related to the numbers in pi, if at all.

    I couldn't find much of anything online.

    I just found something else slightly interesting about the series of repunits 111 through 999 and pi

    The total of the odd repunits is 2775
    The total of the even repunits is 2220

    The total of both is 4995, all three numbers are figurates ( no surprise, there are lots of figurates )

    Following the Sn→Pn→Sn rule for 2220 and 4995, the two series end up joining at 890, where:

    2220 → 9334 → 214 → 102 → 163 → 1410 → 7766 → 890
    4995 → 776 → 890

    Not really that odd, as most of the numbers seem to eventually join to the same series following that rule.
     
  7. Interesting

    A friend of mine wrote a javascript program to search for loops ( I see they are also referred to as " orbits " ) with the first 1000 numbers, there were 88, iirc, looping numbers, each composed of a series of looping numbers, some containing rather large values.

    127,494 was the largest number as of yet.
     
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