# Thread: A slice of pi

1. ## A slice of pi

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

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

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...

--------

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

---------

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

Cheers

2. Originally Posted by Snakeboy
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

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

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...

--------

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

---------

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

Cheers
Hey there Snakeboy,

It's an interesting coincidence that you bring up loops in pi because I wrote a little program yesterday to confirm your previous observations about pi, and for fun I modified it to look for loops in the first million digits. But your comments do not seem entirely accurate. The numbers don't cycle or converge like you say they do. I have only found a few numbers that cycle even when looking in 200 million digits of pi. A good example of a cycle starts with 37:

37 => 46 => 19 => 37

And here is another cycle that begins with the number 40:

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

The number 8 does not cycle in the first 200 million digits of pi. I used this page to check the results that were larger than a million (which is all my little program could handle). Here is how it begins:

8 => 11 => 94 => 58 => 10 => 49 => 57 => 404 => 1272 => 8699 => 3292 => 3332 => 48033 => 90311 => 112817 => 1149731 => 24909936 => 1838500 => 5264650 => 29232231 => 76236585 => 64535680 => 49758988 => 191873638 => not found

The "not found" means that it the number 191873638 does not appear in the first 200 million digits of pi. It did not "converge" to anything. It did not cycle. The sequence could continue increasing forever for all we know.

I think you mistake is to use the word "converge." In mathematics, that word means that a sequence gets closer and closer to a single value. For example, consider the sequence defined by ratios of Fibonacci numbers:

R(n) = F(n)/F(n-1)

This sequence converges to the Golden Ratio: R(n) => Phi as n => infinity. It begins 1, 2, 1.5, 1.666..., 1.6, 1.625, etc.

Now I do understand what you are getting at when you say that they "converge" - what you mean is that two sequences beginning with different starting numbers merge and become the same sequence, whether or not the sequence diverges to infinity or loops. I don't find that particularly significant because the you were just looking at single digits, and they are first found near the beginning so their place numbers would be expected to be small, and hence coincide with small integers.

In any case, math is fun. Thanks for sharing your research.

Richard

3. Sweet, I figured you'd catch anything, it's good you caught that early, it seemed odd, lol.

Anyway, I'm up to 20 so far, 16 seems to go to 40 and loop back, and 19 seems to loop on 37

You ever see how the Fibonacci numbers can all be reduced down to the same 24 digit string ?

I thought that was pretty cool, read about it while trying to learn some series analysis.

4. aha, I just realized the search window on my pi search page seems to randomly count a space as a digit of some sort

I'll have to try your page

5. Well, this does seem interesting, excluding my hare-brained mistakes right off the bat

Is it safe to say that all numbers within a loop are themselves looping numbers ?

Like the 40 to 40 loop you gave as an example.

6. So I went through the first 50 numbers, double checked everything, so I hope this is correct

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

Something neat, I thought, :

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 numbers down the line:

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

Further down the number line 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

7. Originally Posted by Snakeboy
Well, this does seem interesting, excluding my hare-brained mistakes right off the bat

Is it safe to say that all numbers within a loop are themselves looping numbers ?

Like the 40 to 40 loop you gave as an example.
Yes, any number that is part of or leads to a loop could be called a "looping" numbers. The other numbers would be divergent because they lead to a sequence that ultimately goes to infinity. I doubt there is any way to prove that any numbers ultimately diverge, since that would probably require knowing all the digits of pi. And it seems unlikely that even most numbers loop because the density of the digits smaller than "place number n" keeps getting thinner.

To understand this, consider how many distinct 2 digit numbers we could fit in the first hundred digits of pi:

x1 (i.e 11, 21, 31, etc.)
x2 (i.e. 12, 22, 32, etc.)
x3 (i.e. 13, 23, 33, etc.)

The problem is that we are concatenating them and we only have a string of 100 digits to work with.

x0: 102030405060708090
x1: 11213141516171819
x2: 12232425262728292
x3: etc.

As you can see, each line requires about 18 digits to cover all the possibilities. But there are ten lines. So that means we need about 180 digits, but we only have 100, so about 45% of the two digit starting numbers won't be found in the first hundred digits, in which case the position number will be larger than the starting number and the sequence will diverge. So I predict that cycles will be pretty rare in the digits of pi, or any sequence of random numbers. With a little work this could be written as a formal theorem.

8. Originally Posted by Snakeboy
So I went through the first 50 numbers, double checked everything, so I hope this is correct

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

Something neat, I thought, :

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 numbers down the line:

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

Further down the number line 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
So you are doing this all by hand, putting the numbers in one at a time? You should learn a little javascript programming. It does all the work for you. Here are all the looping numbers less than a thousand found in the first million digits of pi. (There may be more if we used a larger set of pi digits.) It took about ten minutes to write, and 5 seconds to run.

----------------
Start: 1
1
----------------
Start: 14
1
1
----------------
Start: 16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 19
37
46
19
----------------
Start: 21
93
14
1
1
----------------
Start: 23
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 37
46
19
37
----------------
Start: 39
43
23
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 43
23
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 45
60
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 46
19
37
46
----------------
Start: 60
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 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
40
----------------
Start: 70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
----------------
Start: 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
40
----------------
Start: 73
299
2643
21
93
14
1
1
----------------
Start: 93
14
1
1
----------------
Start: 96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
----------------
Start: 115
921
422
1839
9323
14
1
1
----------------
Start: 127
297
737
299
2643
21
93
14
1
1
----------------
Start: 141
1
1
----------------
Start: 165
238
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 169
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
----------------
Start: 180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
180
----------------
Start: 183
490
907
542
700
306
115
921
422
1839
9323
14
1
1
----------------
Start: 197
37
46
19
37
----------------
Start: 211
93
14
1
1
----------------
Start: 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
40
----------------
Start: 226
964
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
180
----------------
Start: 238
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 264
21
93
14
1
1
----------------
Start: 270
165
238
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 274
462
19
37
46
19
----------------
Start: 286
73
299
2643
21
93
14
1
1
----------------
Start: 297
737
299
2643
21
93
14
1
1
----------------
Start: 299
2643
21
93
14
1
1
----------------
Start: 306
115
921
422
1839
9323
14
1
1
----------------
Start: 326
274
462
19
37
46
19
----------------
Start: 333
1698
25318
33479
45671
88095
40332
4592
60
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 356
615
1029
8196
197
37
46
19
37
----------------
Start: 375
46
19
37
46
----------------
Start: 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
40
----------------
Start: 399
43
23
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 406
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
----------------
Start: 422
1839
9323
14
1
1
----------------
Start: 433
23
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 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
40
----------------
Start: 459
60
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 462
19
37
46
19
----------------
Start: 464
1159
921
422
1839
9323
14
1
1
----------------
Start: 490
907
542
700
306
115
921
422
1839
9323
14
1
1
----------------
Start: 495
464
1159
921
422
1839
9323
14
1
1
----------------
Start: 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
40
----------------
Start: 514
2117
93
14
1
1
----------------
Start: 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
40
----------------
Start: 542
700
306
115
921
422
1839
9323
14
1
1
----------------
Start: 545
4338
23
16
40
70
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
----------------
Start: 557
1101
2779
15008
15375
202762
107788
780706
96071
53594
141
1
1
----------------
Start: 607
286
73
299
2643
21
93
14
1
1
----------------
Start: 609
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 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
40
----------------
Start: 615
1029
8196
197
37
46
19
37
----------------
Start: 648
226
964
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
180
----------------
Start: 656
514
2117
93
14
1
1
----------------
Start: 665
211
93
14
1
1
----------------
Start: 700
306
115
921
422
1839
9323
14
1
1
----------------
Start: 706
96
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
----------------
Start: 714
609
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 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
40
----------------
Start: 737
299
2643
21
93
14
1
1
----------------
Start: 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
40
----------------
Start: 819
197
37
46
19
37
----------------
Start: 872
648
226
964
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
180
----------------
Start: 874
1949
495
464
1159
921
422
1839
9323
14
1
1
----------------
Start: 902
714
609
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 903
356
615
1029
8196
197
37
46
19
37
----------------
Start: 907
542
700
306
115
921
422
1839
9323
14
1
1
----------------
Start: 921
422
1839
9323
14
1
1
----------------
Start: 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
40
----------------
Start: 932
14
1
1
----------------
Start: 937
45
60
127
297
737
299
2643
21
93
14
1
1
----------------
Start: 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
40
----------------
Start: 964
180
3664
24717
15492
84198
65489
3725
16974
41702
3788
5757
1958
14609
62892
44745
9385
169
40
70
96
180
----------------
Start: 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
40
----------------
Start: 996
459
60
127
297
737
299
2643
21
93
14
1
1

9. Impressive, Richard.

I have been doing this by hand, unfortunately, also cross-referencing them against some other things, so it's rather tedious.

I figured for a really complex analysis it would be a no-brainer to have a program do the looking, I unfortunately have not programmed since learning Basic on the Apple IIe

I've not been able to find too much work in this area, aside from a little bit on the OEIS

http://oeis.org/A232013

Apparently these are called " orbits ", seems fitting.

10. Happy Pi day !

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