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Digital Invariants
This is a little known oddity of the numbers, I've only found a handful of references, shown to me by a Professor, Dr. Tim Flood. The idea is this, take the sum of the squares of the digits of any number and by repeating this process a few times, you will retrieve a 1 or 58 everytime. Try 128:
128 => 12 + 22 + 82 = 69
69 => 62 + 92 = 117
117 => 12 + 12 + 72 = 51
51 => 52 + 12 = 26
26 => 22 + 62 = 40
40 => 42 + 02 = 16
16 => 12 + 62 = 37
37 => 32 + 72 = 58
58 is not unique, it is part of a larger loop system. The entire loop consists of 8 points, called Digital Invariants (DI). And we name each loop by its Least Digital Invariant (LDI). Thus second power 4 loop contains 58 and looks like:
4 => 16 => 37 => 58 => 89 => 42 => 20 => 4
And we can see where we fell into the loop in our example of 128, when 26 => 40 which is 4 for our proposes since we are working with the only the digits.
The third power follows the same process, thus we can obtained a loop, 55 is a third power LDI.
64 => 63 + 43 = 280
280 => 23 + 83 + 03 = 520
520 => 53 + 23 + 03 = 133 *
133 => 13 + 33 + 33 = 55 *
55 => 53 + 53 = 250 *
250 => 23 + 53 + 0 = 133 *
The third power has 15 digital invariants:
1,55,133,136,153,160,217,244,250,352,370,371,407,919,1459
which produce 9 third power loops:
1 => 1
55 => 250 => 133 => 55
136 => 244 => 136
153 => 153
160 => 217 => 352 => 160
370 => 370
371 => 371
407 => 407
919 => 1459 => 919
Here we note the two different classes of LDIs. An LDI that results in a loop of length one is called a Perfect Digital Invariant (PDI), else the LDI is referred to as a Recurring Digital Invariant (RDI).
The process works for all other powers, a proof is given later. I have gathered data for the first 17 powers which produced a table with some of the more interesting highlights:
Power |
DI |
Loop |
RDI |
PDI |
Longest Loop |
Average Loop Size |
Largest LDI |
1 |
9 |
9 |
0 |
9 |
1 |
1 |
9* |
2 |
9 |
2 |
1 |
1 |
8 |
4.500 |
4 |
3 |
15 |
9 |
4 |
5 |
3 |
1.667 |
919 |
4 |
13 |
6 |
2 |
4 |
7 |
2.167 |
9474* |
5 |
103 |
16 |
9 |
7 |
28 |
6.438 |
194979* |
6 |
51 |
7 |
5 |
2 |
30 |
7.286 |
548834* |
7 |
271 |
17 |
11 |
6 |
92 |
15.941 |
14459929* |
8 |
186 |
7 |
3 |
4 |
154 |
26.571 |
88593477* |
9 |
321 |
19 |
14 |
5 |
93 |
16.895 |
912985159* |
10 |
238 |
8 |
6 |
2 |
123 |
29.750 |
4679307774* |
11 |
586 |
24 |
15 |
9 |
181 |
24.417 |
94204591914 |
12 |
264 |
5 |
4 |
1 |
133 |
52.800 |
98840282759 |
13 |
345 |
14 |
12 |
2 |
146 |
24.642 |
3656948275943 |
14 |
547 |
13 |
11 |
2 |
381 |
42.077 |
28116440335967* |
15 |
728 |
12 |
11 |
1 |
362 |
60.667 |
255349823145519 |
16 |
564 |
7 |
4 |
3 |
381 |
80.571 |
4338281769391371* |
17 |
1364 |
22 |
17 |
5 |
528 |
62.000 |
35875699062250035* |
*PDI
Dr. Tim Flood and I have produced a proof to the existence of this loops for all higher powers, the view this proof click here -- DI Proof
There have been many interesting questions asked about this subject. The most prevalent is if/ how the process would work in bases other than base 10. I believe it would work similarly but that is an exercise left for another day. Another question is which numbers go to 1? This process could be used in classroom discovery projects and is easily accessible to students from upper high school into college. There are many other directions to be followed within this research, and I hope others will pursues them. As I plan to do so myself.
Bibliography:
Math Menagerie by Robert R. Kadesch, Harper & Row, New York, 1970
Mathematics on Vacation by Joseph S. Madachy, Charles Scribner's Sons, New York, 1966
"On a Connection Between Programming and Mathematics," by Jozef Hvorecky in SIGSCE Bulletin, 22:4, December, 1990
Unsolved Problems in Number Theory, 2nd Ed., by Richard K. Guy, Springer-Verlag, 1994.
And I would like to add a special thank
you to Dr. Tim Flood for all the work he did with me to complete
the project and prepare for conferences.
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