perfect digit-to-digit invariant

In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number{{cite arXiv|last=van Berkel|first=Daan|title=On a curious property of 3435|year=2009|eprint=0911.3038|class=math.HO }}) is a natural number in a given number base $b$ that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because $3435 = 3^3 + 4^4 + 3^3 + 5^5$ . The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009,Olry, Regis and Duane E. Haines. [https://books.google.com/books?id=h30pAgAAQBAJ&pg=PA136 "Historical and Literary Roots of Münchhausen Syndromes"], from Literature, Neurology, and Neuroscience: Neurological and Psychiatric Disorders, Stanley Finger, Francois Boller, Anne Stiles, eds. Elsevier, 2013. p.136. as this evokes the story of Baron Munchausen raising himself up by his own ponytail because each digit is raised to the power of itself.Daan van Berkel, [https://arxiv.org/abs/0911.3038 On a curious property of 3435.]{{cite book|last1=Parker|first1=Matt|title=Things to Make and Do in the Fourth Dimension|date=2014|publisher=Penguin UK|isbn=9781846147654|page=28|url=https://books.google.com/books?id=AOu2AwAAQBAJ&pg=PT28|access-date=2 May 2015}}

Definition

Let $n$ be a natural number which can be written in base $b$ as the k-digit number $d_{k-1}d_{k-2}...d_{1}d_{0}$ where each digit $d_i$ is between $0$ and $b-1$ inclusive, and $n = \sum_{i=0}^{k-1} d_{i}b^{i}$ . We define the function $F_b : \mathbb{N} \rightarrow \mathbb{N}$ as $F_b(n) = \sum_{i=0}^{k - 1} {d_i}^{d_i}$ . (As 0⁰ is usually undefined, there are typically two conventions used, one where it is taken to be equal to one, and another where it is taken to be equal to zero.[http://www.magic-squares.net/narciss.htm Narcisstic Number], Harvey Heinz{{cite book

| last = Wells

| first = David

| title = The Penguin Dictionary of Curious and Interesting Numbers

| publisher = Penguin

| location = London

| year = 1997

| page = 185

| isbn = 0-14-026149-4}}) A natural number $n$ is defined to be a perfect digit-to-digit invariant in base b if $F_b(n) = n$ . For example, the number 3435 is a perfect digit-to-digit invariant in base 10 because $3^3 + 4^4 + 3^3 + 5^5 = 27 + 256 + 27 + 3125 = 3435$ .

$F_b(1) = 1$ for all $b$ , and thus 1 is a trivial perfect digit-to-digit invariant in all bases, and all other perfect digit-to-digit invariants are nontrivial. For the second convention where $0^0 = 0$ , both $0$ and $1$ are trivial perfect digit-to-digit invariants.

A natural number $n$ is a sociable digit-to-digit invariant if it is a periodic point for $F_{b}$ , where $F_{b}^k(n) = n$ for a positive integer $k$ , and forms a cycle of period $k$ . A perfect digit-to-digit invariant is a sociable digit-to-digit invariant with $k = 1$ . An amicable digit-to-digit invariant is a sociable digit-to-digit invariant with $k = 2$ .

All natural numbers $n$ are preperiodic points for $F_b$ , regardless of the base. This is because all natural numbers of base $b$ with $k$ digits satisfy $b^{k-1} \leq n \leq (k){(b - 1)}^{b-1}$ . However, when $k \geq b+1$ , then $b^{k-1} > (k){(b - 1)}^{b-1}$ , so any $n$ will satisfy $n > F_b(n)$ until $n < b^{b+1}$ . There are a finite number of natural numbers less than $b^{b+1}$ , so the number is guaranteed to reach a periodic point or a fixed point less than $b^{b+1}$ , making it a preperiodic point. This means also that there are a finite number of perfect digit-to-digit invariant and cycles for any given base $b$ .

The number of iterations $i$ needed for $F_b^{i}(n)$ to reach a fixed point is the $b$ -factorion function's persistence of $n$ , and undefined if it never reaches a fixed point.

Perfect digit-to-digit invariants and cycles of Fb for specific b

All numbers are represented in base $b$ .

=Convention 00 = 1 =

class="wikitable"
Base ! Nontrivial perfect digit-to-digit invariants ( $n \neq 1$ ) ! Cycles
2 \| 10 \| $\varnothing$
3 \| 12, 22 \| 2 → 11 → 2
4 \| 131, 313 \| 2 → 10 → 2
5 \| $\varnothing$ \| 2 → 4 → 2011 → 12 → 10 → 2 104 → 2013 → 113 → 104
6 \| 22352, 23452 \| 4 → 1104 → 1111 → 4 23445 → 24552 → 50054 → 50044 → 24503 → 23445
7 \| 13454 \| 12066 → 536031 → 265204 → 265623 → 551155 → 51310 → 12125 → 12066
8 \| \| 405 → 6466 → 421700 → 3110776 → 6354114 → 142222 → 421 → 405
9 \| 31, 156262, 1656547 \|
10 \| 3435 \|
11 \| \|
12 \| 3A67A54832 \|

class="wikitable"

Base

! Nontrivial perfect digit-to-digit invariants ( $n \neq 1$ )

! Cycles

| 10

| $\varnothing$

| 12, 22

| 2 → 11 → 2

| 131, 313

| 2 → 10 → 2

| $\varnothing$

2 → 4 → 2011 → 12 → 10 → 2

104 → 2013 → 113 → 104

| 22352, 23452

4 → 1104 → 1111 → 4

23445 → 24552 → 50054 → 50044 → 24503 → 23445

| 13454

| 12066 → 536031 → 265204 → 265623 → 551155 → 51310 → 12125 → 12066

| 405 → 6466 → 421700 → 3110776 → 6354114 → 142222 → 421 → 405

| 31, 156262, 1656547

| 3435

| 3A67A54832

=Convention 00 = 0=

class="wikitable"
Base ! Nontrivial perfect digit-to-digit invariants ( $n \neq 0$ , $n \neq 1$ ) ! Cycles
2 \| $\varnothing$ \| $\varnothing$
3 \| 12, 22 \| 2 → 11 → 2
4 \| 130, 131, 313 \| $\varnothing$
5 \| 103, 2024 \| 2 → 4 → 2011 → 11 → 2 9 → 2012 → 9
6 \| 22352, 23452 \| 5 → 22245 → 23413 → 1243 → 1200 → 5 53 → 22332 → 150 → 22250 → 22305 → 22344 → 2311 → 53
7 \| 13454 \|
8 \| 400, 401 \|
9 \| 30, 31, 156262, 1647063, 1656547, 34664084 \|
10 \| 3435, 438579088 \|
11 \| $\varnothing$ \| $\varnothing$
12 \| 3A67A54832 \|

class="wikitable"

Base

! Nontrivial perfect digit-to-digit invariants ( $n \neq 0$ , $n \neq 1$ )

! Cycles

| $\varnothing$

| 12, 22

| 2 → 11 → 2

| 130, 131, 313

| $\varnothing$

| 103, 2024

2 → 4 → 2011 → 11 → 2

9 → 2012 → 9

| 22352, 23452

5 → 22245 → 23413 → 1243 → 1200 → 5

53 → 22332 → 150 → 22250 → 22305 → 22344 → 2311 → 53

| 13454

| 400, 401

| 30, 31, 156262, 1647063, 1656547, 34664084

| 3435, 438579088

| $\varnothing$

| 3A67A54832

Programming examples

The following program in Python determines whether an integer number is a Munchausen Number / Perfect Digit to Digit Invariant or not, following the convention $0^0 = 1$ .

num = int(input("Enter number:"))

temp = num

s = 0.0

while num > 0:

digit = num % 10

num //= 10

s+= pow(digit, digit)

if s == temp:

print("Munchausen Number")

else:

print("Not Munchausen Number")

The examples below implement the perfect digit-to-digit invariant function described in the definition above to search for perfect digit-to-digit invariants and cycles in Python for the two conventions.

= Convention 00 = 1 =

def pddif(x: int, b: int) -> int:

total = 0

while x > 0:

total = total + pow(x % b, x % b)

x = x // b

return total

def pddif_cycle(x: int, b: int) -> list[int]:

seen = []

while x not in seen:

seen.append(x)

x = pddif(x, b)

cycle = []

while x not in cycle:

cycle.append(x)

x = pddif(x, b)

return cycle

= Convention 00 = 0 =

def pddif(x: int, b: int) -> int:

total = 0

while x > 0:

if x % b > 0:

total = total + pow(x % b, x % b)

x = x // b

return total

def pddif_cycle(x: int, b: int) -> list[int]:

seen = []

while x not in seen:

seen.append(x)

x = pddif(x, b)

cycle = []

while x not in cycle:

cycle.append(x)

x = pddif(x, b)

return cycle

References

External links

{{cite web|last=Parker|first=Matt|title=3435|url=http://www.numberphile.com/videos/3435.html|work=Numberphile|publisher=Brady Haran|access-date=2013-04-01|archive-url=https://web.archive.org/web/20170413072736/http://www.numberphile.com/videos/3435.html|archive-date=2017-04-13|url-status=dead}}

Category:Arithmetic dynamics

Category:Base-dependent integer sequences

Category:Articles with example Python (programming language) code

perfect digit-to-digit invariant

Definition

Perfect digit-to-digit invariants and cycles of F<sub>b</sub> for specific b

=Convention 0<sup>0</sup> = 1 =

=Convention 0<sup>0</sup> = 0=

Programming examples

= Convention 0<sup>0</sup> = 1 =

= Convention 0<sup>0</sup> = 0 =

See also

References

External links