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

Let be a natural number. We define the perfect digital invariant function (also known as a happy function, from happy numbers) for base and power to be the following:

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]

Perfect digital invariants of F2,b

By definition, any three-digit perfect digital invariant for with natural number digits , , has to satisfy the cubic Diophantine equation . However, 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

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

Perfect digital invariants of F3,b

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 . However, 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

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
14130131203
27250251305
310370371407
413490491509
5165B05B160B
6196D06D170D
7227F07F180F
8258H08H190H
9289J09J1A0J

b = 3k + 2

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
15103
28205
311307
414409
51750B
62060D
72370F
82680H
92990J

b = 6k + 4

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
04021
110153
216285
3223B7
4284E9

Perfect digital invariants and cycles of Fp,b for specific p and b

All numbers are represented in base .

Nontrivial perfect digital invariants Cycles
2312, 222 → 11 → 2
4
523, 334 → 31 → 20 → 4
65 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5
713, 34, 44, 632 → 4 → 22 → 11 → 2

16 → 52 → 41 → 23 → 16

824, 64

4 → 20 → 4

5 → 31 → 12 → 5

15 → 32 → 15

945, 55

58 → 108 → 72 → 58

75 → 82 → 75

104 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4
1156, 66

5 → 23 → 12 → 5

68 → 91 → 75 → 68

1225, A5

5 → 21 → 5

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

18 → 55 → 42 → 18

68 → 84 → 68

1314, 36, 67, 77, A6, C428 → 53 → 28

79 → A0 → 79

98 → B2 → 98

141B → 8A → BA → 11B → 8B → D3 → CA → 136 → 34 → 1B

29 → 61 → 29

1578, 882 → 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

16D → A9 → B5 → 92 → 55 → 32 → D
331222 → 22 → 121 → 101 → 2
420, 21, 130, 131, 203, 223, 313, 332
5103, 43314 → 230 → 120 → 14
6243, 514, 105513 → 44 → 332 → 142 → 201 → 13
712, 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

8134, 205, 463, 660, 661662 → 670 → 1057 → 725 → 734 → 662
930, 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

10153, 370, 371, 407

55 → 250 → 133 → 55

136 → 244 → 136

160 → 217 → 352 → 160

919 → 1459 → 919

1132, 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

12577, 668, A83, 11AA
13490, 491, 509, B8513 → 22 → 13
14136, 409
15C3A, D87
1623, 40, 41, 156, 173, 208, 248, 285, 4A5, 580, 581, 60B, 64B, 8C0, 8C1, 99A, AA9, AC3, CA8, E69, EA0, EA1
43

121 → 200 → 121

122 → 1020 → 122

41103, 33033 → 1101 → 3
52124, 2403, 3134

1234 → 2404 → 4103 → 2323 → 1234

2324 → 2434 → 4414 → 11034 → 2324

3444 → 11344 → 4340 → 4333 → 3444

6
7
820, 21, 400, 401, 420, 421
9432, 2466
531020, 1021, 2102, 10121
4200

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

3311 → 13220 → 10310 → 3311

Extension to negative integers

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

Balanced ternary

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

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

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

References

  1. 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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