There are some interesting loops in pi digits, thought I'd toss this out there for funsies

Excuse my novice notation, haha, Richard, if you have a more elegant way to write this out, please do

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, I started with the numbers 0 through 9, and located the number 0, the position of the 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 0, 2 through 7, and 9, the series converge on each other fairly quick, 1 is a self-locating digit in pi ( a rarity in itself )

However, at 8 something neat happens:

As the series Sn → Pn → Pn progresses for 8, at the 12th step, you come to 3332, and progressing the series, it loops back on itself to 3332 after 11 steps.

The string 3332 occurs at position 48033, following the rule, it comes back to 3332, ( → ∞ )

I have not been able to find where and when the numbers actually loop back to themselves, if they do at all, However for the number 8 it seems impossible, following that rule of Sn → Pn → Sn :O

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There are some other loops, here's one:

The number 169, located at the 40th place, following Sn → Pn → Pn , loops back to 196 after 21 steps in the series

So, if you like:

1. Find when and where the numbers 0 - 7, and 9 loop on themselves in pi digits following the rule defined above, or, state why they cannot. ( I so far have not found them to loop )

2. There are two numbers we cannot do this with, 1 and 8

Explain why it's impossible with the number 8

3. Find a proof of whether or not there are an infinite amount of looping numbers in pi

Here are the numbers 0 through 9, following the rule:

0 )→ 32 → 15 → 3 → 9 → 5 → 4 → 2 → 6 → 7 → 13...

1 )→ 1 → ∞

2 )→ 6 → 7 → 13 → ...

3 )→ 9 → 5 → 4 → 2 → 6 → 7 → 13...

4 )→ 2 → 6 → 7 → 13...

5 )→ 4 → 2 → 6 → 7 → 13...

6 )→ 7 → 13 → ...

7 )→ 13 → ...

8 )→ 11 → 94 → 58 → 10 → 49 → 57 → 404 → 1272 → 8699 → 3292 → 3332 → 48033 → 90311 → 7573 → 1959 → 985 → 2154 → 5276 → 33991 → 18316 → 32928 → 3332 → ∞

9 )→ 5 → 4 → 2 → 6 → 7 → 13...

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8 of the series converge at 13

8 of the series converge at 7

7 of the series converge at 6

4 of the series converge on 5

4 of the series converge on 4

6 of the series converge on 2

For 0 and 2 -9 , the series grow fast and and the strings show no signs of shrinking back towards their starting values

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Have fun, and if you can simplify the notation or find any more looping numbers, please add them

Cheers

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