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Loops within Pi

Having watched the Numberphile video "Strings and Loops within Pi" - https://www.youtube.com/watch?v=W20aT14t8Pw , I wrote a program to analyse the first 1,000,000 digits of Pi. As I follow-up I wrote a program to analyse the first 1,000,000,000 digits.

Here are the complete loops I found between < 1000 in the first 1,000,000,000 digits:

  • Num 19 Len: 3 Ans: 19->37->46->19
  • Num: 37 Len: 3 Ans: 37->46->19->37
  • Num: 40 Len: 20 Ans: 40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40
  • Num: 46 Len: 3 Ans: 46->19->37->46
  • Num: 70 Len: 20 Ans: 70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40->70
  • Num: 96 Len: 20 Ans: 96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40->70->96
  • Num: 169 Len: 20 Ans: 169->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169
  • Num: 180 Len: 20 Ans: 180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40->70->96->180

Note that 40, 70, 96, 168, and 180 are all the same chain with just different start points and end points. 19, 37, and 46 are also a complete chain.

Current limits and a risky hypothesis

Lowest

  • Num 19 Len: 3 Ans: 19->37->46->19

Highest

  • Num: 84198 Len: 20 Ans: 84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40->70->96->180->3664->24717->15492->84198

Observations

Note that 84198 is the highest number in any of the loops. Also note that besides 19, 47, and 36, all loops go through ->40->70->96->

This means that the loop for 169 and the loop for 84198 are the same loop, just shifted with a different start and end place.

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

Risky Hypothesis

Here is a risky hypothesis: There are only two chains in Pi < 711939213.

  • 46->19->37->46
  • 169->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169

Picking 711939213 is partially arbitrary since it is a loop terminator like 1 and not a loop like 169->40->70->96->180->3664...

More details on 711939213 below.

Infinite loops

From this I also noted that some loops are infinite, in that they loop back into a previous part of the sequence and therefore loop around and around without an exit condition, an infinite loop.

For example:

819->197->37->46->19->37->...
  • 819 occurs at position 197 (counting from the first digit after the decimal point, the 3. is not counted).
  • 197 occurs at position 37.
  • 37 occurs at position 46.
  • 46 occurs at position 19.
  • 19 occurs at position 37.
  • 37 occurs at position 46.
  • 46 occurs at position 19.
  • 19 occurs at position 37.
  • ...

Here are the others < 1000:

  • Num: 16 Len: 21 Ans: 16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 23 Len: 22 Ans: 23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 39 Len: 24 Ans: 39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 43 Len: 23 Ans: 43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 61 Len: 27 Ans: 61->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 71 Len: 25 Ans: 71->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 165 Len: 23 Ans: 165->238->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 197 Len: 4 Ans: 197->37->46->19-> Infinite loop: 37
  • Num: 219 Len: 26 Ans: 219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 238 Len: 22 Ans: 238->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 270 Len: 24 Ans: 270->165->238->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 356 Len: 8 Ans: 356->615->1029->8196->197->37->46->19-> Infinite loop: 37
  • Num: 375 Len: 4 Ans: 375->46->19->37-> Infinite loop: 46
  • Num: 394 Len: 29 Ans: 394->526->612->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 399 Len: 24 Ans: 399->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 406 Len: 21 Ans: 406->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40-> Infinite loop: 70
  • Num: 433 Len: 23 Ans: 433->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 440 Len: 31 Ans: 440->511->394->526->612->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 511 Len: 30 Ans: 511->394->526->612->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 526 Len: 28 Ans: 526->612->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 545 Len: 24 Ans: 545->4338->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 612 Len: 27 Ans: 612->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 615 Len: 7 Ans: 615->1029->8196->197->37->46->19-> Infinite loop: 37
  • Num: 706 Len: 21 Ans: 706->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40->70-> Infinite loop: 96
  • Num: 716 Len: 25 Ans: 716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 806 Len: 34 Ans: 806->967->1400->4365->11353->127494->4220->2371->1925->1166->3993->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 819 Len: 5 Ans: 819->197->37->46->19-> Infinite loop: 37
  • Num: 903 Len: 9 Ans: 903->356->615->1029->8196->197->37->46->19-> Infinite loop: 37
  • Num: 931 Len: 32 Ans: 931->440->511->394->526->612->219->716->39->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 943 Len: 25 Ans: 943->399->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40
  • Num: 967 Len: 33 Ans: 967->1400->4365->11353->127494->4220->2371->1925->1166->3993->43->23->16->40->70->96->180->3664->24717->15492->84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169-> Infinite loop: 40

Analysis

Looking at 197 we see:197->37->46->19-> Infinite loop: 37. Since 19 is a closed loop of 19->37->46->19 and that 7 is the next digit after 19 (at position 37), then 197 starts in the same way, but loops back to 19, not to 197 and is therefore an infinite loop.

Difference between the 1,000,000 digit search and the 1,000,000,000 digit search

Pi is loaded into memory for the 1,000,000 digit search, whereas it remains on disk for the 1,000,000,000 digit search. The 1,000,000 is faster because of the size of the search set and the difference in speed between memory and disk.

Searching 1,000,000,000 digitis tends not to provide extra results, however it did provide two more infinite loops during the search between 999 and 9999 (namely 3708 and 7299). See links below. Such searches also extend the tail length of unsolved loops. Using 2 as an example:

  • For 1,000,000 digits: 2->6->7->13->110->174->155->314->2120->5360->24671->119546->193002->240820->274454->153700-> Not found: 153700
  • For 1,000,000,000 digits: 2->6->7->13->110->174->155->314->2120->5360->24671->119546->193002->240820->274454->153700->1397287->17916598->26245242->8880928->7320921->14726415->42969065->35308126->14978764->68756682->300921774-> Not found: 300921774

More loops

From time to time I might run my program for a new range, I won't update this page, but rather just do a raw dump of the results and link to it here.

From 999 to 9999

Can be found here: https://github.com/garyexplains/examples/blob/master/loopswithpi-from999to9999.md

One interesting loop in that collection is: 1971->37->46->19-> Infinite loop: 37

1971 is an infinite loop like 197 because 19 is a closed loop.

This would mean that 19716, 197169, 1971693,... are also infinite loops.

From 9999 to 99999

Can be found here: https://github.com/garyexplains/examples/blob/master/loopswithpi-from9999to99999.md

Where we see 19716->37->46->19-> Infinite loop: 37 as described above.

From 99999 to 9999999

Can be found here: https://github.com/garyexplains/examples/blob/master/loopswithpi-from99999to9999999.md

There are no loops in the range 99999 to 9999999 when searching Pi with 1,000,000 digits.

This means that 84198 is the highest loop I have found:

  • Num: 84198 Len: 20 Ans: 84198->65489->3725->16974->41702->3788->5757->1958->14609->62892->44745->9385->169->40->70->96->180->3664->24717->15492->84198

Loops ending in 1

Since 1 is a self-locating number within Pi, when a sequence hits 1 it becomes a loop. Here are the number sequences, < 1000, that end in a 1 loop:

Num: 14 Len: 2 Ans: 14->1-> Infinite loop: 1
Num: 21 Len: 4 Ans: 21->93->14->1-> Infinite loop: 1
Num: 45 Len: 11 Ans: 45->60->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 60 Len: 10 Ans: 60->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 73 Len: 7 Ans: 73->299->2643->21->93->14->1-> Infinite loop: 1
Num: 93 Len: 3 Ans: 93->14->1-> Infinite loop: 1
Num: 115 Len: 7 Ans: 115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 127 Len: 9 Ans: 127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 141 Len: 2 Ans: 141->1-> Infinite loop: 1
Num: 183 Len: 13 Ans: 183->490->907->542->700->306->115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 211 Len: 4 Ans: 211->93->14->1-> Infinite loop: 1
Num: 264 Len: 5 Ans: 264->21->93->14->1-> Infinite loop: 1
Num: 286 Len: 8 Ans: 286->73->299->2643->21->93->14->1-> Infinite loop: 1
Num: 297 Len: 8 Ans: 297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 299 Len: 6 Ans: 299->2643->21->93->14->1-> Infinite loop: 1
Num: 306 Len: 8 Ans: 306->115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 333 Len: 18 Ans: 333->1698->25318->33479->45671->88095->40332->4592->60->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 422 Len: 5 Ans: 422->1839->9323->14->1-> Infinite loop: 1
Num: 459 Len: 11 Ans: 459->60->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 464 Len: 8 Ans: 464->1159->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 490 Len: 12 Ans: 490->907->542->700->306->115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 495 Len: 9 Ans: 495->464->1159->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 514 Len: 5 Ans: 514->2117->93->14->1-> Infinite loop: 1
Num: 542 Len: 10 Ans: 542->700->306->115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 557 Len: 12 Ans: 557->1101->2779->15008->15375->202762->107788->780706->96071->53594->141->1-> Infinite loop: 1
Num: 607 Len: 9 Ans: 607->286->73->299->2643->21->93->14->1-> Infinite loop: 1
Num: 609 Len: 10 Ans: 609->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 656 Len: 6 Ans: 656->514->2117->93->14->1-> Infinite loop: 1
Num: 665 Len: 5 Ans: 665->211->93->14->1-> Infinite loop: 1
Num: 700 Len: 9 Ans: 700->306->115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 714 Len: 11 Ans: 714->609->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 737 Len: 7 Ans: 737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 874 Len: 11 Ans: 874->1949->495->464->1159->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 902 Len: 12 Ans: 902->714->609->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 907 Len: 11 Ans: 907->542->700->306->115->921->422->1839->9323->14->1-> Infinite loop: 1
Num: 921 Len: 6 Ans: 921->422->1839->9323->14->1-> Infinite loop: 1
Num: 932 Len: 3 Ans: 932->14->1-> Infinite loop: 1
Num: 937 Len: 12 Ans: 937->45->60->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1
Num: 996 Len: 12 Ans: 996->459->60->127->297->737->299->2643->21->93->14->1-> Infinite loop: 1

Other observations

Looking at the sequence A057680 "Self-locating strings within Pi: numbers n such that the string n is at position n (after the decimal point) in decimal digits of Pi" - 1, 16470, 44899, 79873884, 711939213, 36541622473, 45677255610, 62644957128, 656430109694

Ignoring 1, we see that 16470, 44899 and 79873884 all have occurances before their self-locating version. 16470 appears at 1602 before it appears at 16470. This means it can't terminate a loop like 1. Likewise 44899 appears at 13714, 15399, and 41604 before it appears at 44899. So it can not terminate a loop like 1. Also, 79873884 occurs at position 46267046 before it appears at 79873884. So it can't terminate a loop like 1.

However, 711939213 first appears at 711939213. This means it has the potential to terminate a loop like 1. This means that 7119392130, 71193921308, 711939213085, 7119392130851, 71193921308513, 711939213085135, 7119392130851350, ... will all point to 711939213 and therefore terminate.

Note that, 711939213 does not appear again in the first 1,000,000,000 digits of Pi. There is scope here for further searching on a bigger dataset.

Other resources

You can check my findings manually here: https://www.angio.net/pi/ or here https://mathigon.org/course/circles/introduction#pi-digits

Note that https://mathigon.org/course/circles/introduction#pi-digits indexes from 0, not 1. My results are based on an index of 1.

My Code

I want to write some more variants of my code, but in due time I will release the code here.