Perfect digital invariant

In number theory, a perfect digital invariant (PDI) is a number in a given number base () that is the sum of its own digits each raised to a given power ().[1][2]

Definition

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Let   be a natural number. The perfect digital invariant function (also known as a happy function, from happy numbers) for base   and power     is defined as:

 

where   is the number of digits in the number in base  , and

 

is the value of each digit of the number. A natural number   is a perfect digital invariant if it is a fixed point for  , which occurs if  .   and   are trivial perfect digital invariants for all   and  , all other perfect digital invariants are nontrivial perfect digital invariants.

For example, the number 4150 in base   is a perfect digital invariant with  , because  .

A natural number   is a sociable digital invariant if it is a periodic point for  , where   for a positive integer   (here   is the  th iterate of  ), and forms a cycle of period  . A perfect digital invariant is a sociable digital invariant with  , and a amicable digital invariant is a sociable digital invariant with  .

All natural numbers   are preperiodic points for  , regardless of the base. This is because if  ,  , so any   will satisfy   until  . There are a finite number of natural numbers less than  , so the number is guaranteed to reach a periodic point or a fixed point less than  , making it a preperiodic point.

Numbers in base   lead to fixed or periodic points of numbers  .

Proof

If  , then the   bound can be reduced. Let   be the number for which the sum of squares of digits is largest among the numbers less than  .

 
  because  

Let   be the number for which the sum of squares of digits is largest among the numbers less than  .

 
  because  

Let   be the number for which the sum of squares of digits is largest among the numbers less than  .

 
 

Let   be the number for which the sum of squares of digits is largest among the numbers less than  .

 
 

 . Thus, numbers in base   lead to cycles or fixed points of numbers  .

The number of iterations   needed for   to reach a fixed point is the perfect digital invariant function's persistence of  , and undefined if it never reaches a fixed point.

  is the digit sum. The only perfect digital invariants are the single-digit numbers in base  , and there are no periodic points with prime period greater than 1.

  reduces to  , as for any power  ,   and  .

For every natural number  , if  ,   and  , then for every natural number  , if  , then  , where   is Euler's totient function.

Proof

Let

 

be a natural number with   digits, where  , and  , where   is a natural number greater than 1.

According to the divisibility rules of base  , if  , then if  , then the digit sum

 

If a digit  , then  . According to Euler's theorem, if  ,  . Thus, if the digit sum  , then  .

Therefore, for any natural number  , if  ,   and  , then for every natural number  , if  , then  .

No upper bound can be determined for the size of perfect digital invariants in a given base and arbitrary power, and it is not currently known whether or not the number of perfect digital invariants for an arbitrary base is finite or infinite.[1]

F2,b

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By definition, any three-digit perfect digital invariant   for   with natural number digits  ,  ,   has to satisfy the cubic Diophantine equation  .   has to be equal to 0 or 1 for any  , because the maximum value   can take is  . As a result, there are actually two related quadratic Diophantine equations to solve:

  when  , and
  when  .

The two-digit natural number   is a perfect digital invariant in base

 

This can be proven by taking the first case, where  , and solving for  . This means that for some values of   and  ,   is not a perfect digital invariant in any base, as   is not a divisor of  . Moreover,  , because if   or  , then  , which contradicts the earlier statement that  .

There are no three-digit perfect digital invariants for  , which can be proven by taking the second case, where  , and letting   and  . Then the Diophantine equation for the three-digit perfect digital invariant becomes

 
 
 
 

  for all values of  . Thus, there are no solutions to the Diophantine equation, and there are no three-digit perfect digital invariants for  .

F3,b

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There are just four numbers, after unity, which are the sums of the cubes of their digits:

 
 
 
 
These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician. (sequence A046197 in the OEIS)
— G. H. Hardy, A Mathematician's Apology

By definition, any four-digit perfect digital invariant   for   with natural number digits  ,  ,  ,   has to satisfy the quartic Diophantine equation  .   has to be equal to 0, 1, 2 for any  , because the maximum value   can take is  . As a result, there are actually three related cubic Diophantine equations to solve

  when  
  when  
  when  

We take the first case, where  .

b = 3k + 1

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Let   be a positive integer and the number base  . Then:

  •   is a perfect digital invariant for   for all  .
Proof

Let the digits of   be  ,  , and  . Then

 

Thus   is a perfect digital invariant for   for all  .

  •   is a perfect digital invariant for   for all  .
Proof

Let the digits of   be  ,  , and  . Then

 

Thus   is a perfect digital invariant for   for all  .

  •   is a perfect digital invariant for   for all  .
Proof

Let the digits of   be  ,  , and  . Then

 

Thus   is a perfect digital invariant for   for all  .

Perfect digital invariants
         
1 4 130 131 203
2 7 250 251 305
3 10 370 371 407
4 13 490 491 509
5 16 5B0 5B1 60B
6 19 6D0 6D1 70D
7 22 7F0 7F1 80F
8 25 8H0 8H1 90H
9 28 9J0 9J1 A0J

b = 3k + 2

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Let   be a positive integer and the number base  . Then:

  •   is a perfect digital invariant for   for all  .
Proof

Let the digits of   be  ,  , and  . Then

 
 
 
 
 
 
 
 
 
 
 
 
 

Thus   is a perfect digital invariant for   for all  .

Perfect digital invariants
     
1 5 103
2 8 205
3 11 307
4 14 409
5 17 50B
6 20 60D
7 23 70F
8 26 80H
9 29 90J

b = 6k + 4

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Let   be a positive integer and the number base  . Then:

  •   is a perfect digital invariant for   for all  .
Proof

Let the digits of   be  ,  , and  . Then

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Thus   is a perfect digital invariant for   for all  .

Perfect digital invariants
     
0 4 021
1 10 153
2 16 285
3 22 3B7
4 28 4E9

Fp,b

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All numbers are represented in base  .

    Nontrivial perfect digital invariants Cycles
2 3 12, 22 2 → 11 → 2
4    
5 23, 33 4 → 31 → 20 → 4
6   5 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5
7 13, 34, 44, 63 2 → 4 → 22 → 11 → 2

16 → 52 → 41 → 23 → 16

8 24, 64

4 → 20 → 4

5 → 31 → 12 → 5

15 → 32 → 15

9 45, 55

58 → 108 → 72 → 58

75 → 82 → 75

10   4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4
11 56, 66

5 → 23 → 12 → 5

68 → 91 → 75 → 68

12 25, A5

5 → 21 → 5

8 → 54 → 35 → 2A → 88 → A8 → 118 → 56 → 51 → 22 → 8

18 → 55 → 42 → 18

68 → 84 → 68

13 14, 36, 67, 77, A6, C4 28 → 53 → 28

79 → A0 → 79

98 → B2 → 98

14   1B → 8A → BA → 11B → 8B → D3 → CA → 136 → 34 → 1B

29 → 61 → 29

15 78, 88 2 → 4 → 11 → 2

8 → 44 → 22 → 8

15 → 1B → 82 → 48 → 55 → 35 → 24 → 15

2B → 85 → 5E → EB → 162 → 2B

4E → E2 → D5 → CE → 17A → A0 → 6A → 91 → 57 → 4E

9A → C1 → 9A

D6 → DA → 12E → D6

16   D → A9 → B5 → 92 → 55 → 32 → D
3 3 122 2 → 22 → 121 → 101 → 2
4 20, 21, 130, 131, 203, 223, 313, 332  
5 103, 433 14 → 230 → 120 → 14
6 243, 514, 1055 13 → 44 → 332 → 142 → 201 → 13
7 12, 22, 250, 251, 305, 505

2 → 11 → 2

13 → 40 → 121 → 13

23 → 50 → 236 → 506 → 665 → 1424 → 254 → 401 → 122 → 23

51 → 240 → 132 → 51

160 → 430 → 160

161 → 431 → 161

466 → 1306 → 466

516 → 666 → 1614 → 552 → 516

8 134, 205, 463, 660, 661 662 → 670 → 1057 → 725 → 734 → 662
9 30, 31, 150, 151, 570, 571, 1388

38 → 658 → 1147 → 504 → 230 → 38

152 → 158 → 778 → 1571 → 572 → 578 → 1308 → 660 → 530 → 178 → 1151 → 152

638 → 1028 → 638

818 → 1358 → 818

10 153, 370, 371, 407

55 → 250 → 133 → 55

136 → 244 → 136

160 → 217 → 352 → 160

919 → 1459 → 919

11 32, 105, 307, 708, 966, A06, A64

3 → 25 → 111 → 3

9 → 603 → 201 → 9

A → 82A → 1162 → 196 → 790 → 895 → 1032 → 33 → 4A → 888 → 1177 → 576 → 5723 → A3 → 8793 → 1210 → A

25A → 940 → 661 → 364 → 25A

366 → 388 → 876 → 894 → A87 → 1437 → 366

49A → 1390 → 629 → 797 → 1077 → 575 → 49A

12 577, 668, A83, 11AA
13 490, 491, 509, B85 13 → 22 → 13
14 136, 409
15 C3A, D87
16 23, 40, 41, 156, 173, 208, 248, 285, 4A5, 580, 581, 60B, 64B, 8C0, 8C1, 99A, AA9, AC3, CA8, E69, EA0, EA1
4 3  

121 → 200 → 121

122 → 1020 → 122

4 1103, 3303 3 → 1101 → 3
5 2124, 2403, 3134

1234 → 2404 → 4103 → 2323 → 1234

2324 → 2434 → 4414 → 11034 → 2324

3444 → 11344 → 4340 → 4333 → 3444

6  
7  
8 20, 21, 400, 401, 420, 421
9 432, 2466
5 3 1020, 1021, 2102, 10121  
4 200

3 → 3303 → 23121 → 10311 → 3312 → 20013 → 10110 → 3

3311 → 13220 → 10310 → 3311

Extension to negative integers

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Perfect digital invariants can be extended to the negative integers by use of a signed-digit representation to represent each integer.

Balanced ternary

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In balanced ternary, the digits are 1, −1 and 0. This results in the following:

  • With odd powers  ,   reduces down to digit sum iteration, as  ,   and  .
  • With even powers  ,   indicates whether the number is even or odd, as the sum of each digit will indicate divisibility by 2 if and only if the sum of digits ends in 0. As   and  , for every pair of digits 1 or −1, their sum is 0 and the sum of their squares is 2.

Relation to happy numbers

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A happy number   for a given base   and a given power   is a preperiodic point for the perfect digital invariant function   such that the  -th iteration of   is equal to the trivial perfect digital invariant  , and an unhappy number is one such that there exists no such  .

Programming example

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The example below implements the perfect digital invariant function described in the definition above to search for perfect digital invariants and cycles in Python. This can be used to find happy numbers.

def pdif(x: int, p: int, b: int) -> int:
    """Perfect digital invariant function."""
    total = 0
    while x > 0:
        total = total + pow(x % b, p)
        x = x // b
    return total

def pdif_cycle(x: int, p: int, b: int) -> list[int]:
    seen = []
    while x not in seen:
        seen.append(x)
        x = pdif(x, p, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = pdif(x, p, b)
    return cycle

See also

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References

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  1. ^ a b Perfect and PluPerfect Digital Invariants Archived 2007-10-10 at the Wayback Machine by Scott Moore
  2. ^ PDIs by Harvey Heinz
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