In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number[1]) is a natural number in a given number base ${\displaystyle b}$ that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because ${\displaystyle 3435=3^{3}+4^{4}+3^{3}+5^{5))$. The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009,[2] 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.[3][4]

Definition

Let ${\displaystyle n}$ be a natural number which can be written in base ${\displaystyle b}$ as the k-digit number ${\displaystyle d_{k-1}d_{k-2}...d_{1}d_{0))$ where each digit ${\displaystyle d_{i))$ is between ${\displaystyle 0}$ and ${\displaystyle b-1}$ inclusive, and ${\displaystyle n=\sum _{i=0}^{k-1}d_{i}b^{i))$. We define the function ${\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }$ as ${\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}{d_{i))^{d_{i))}$. (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.[5][6]) A natural number ${\displaystyle n}$ is defined to be a perfect digit-to-digit invariant in base b if ${\displaystyle F_{b}(n)=n}$. For example, the number 3435 is a perfect digit-to-digit invariant in base 10 because ${\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435}$.

${\displaystyle F_{b}(1)=1}$ for all ${\displaystyle b}$, 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 ${\displaystyle 0^{0}=0}$, both ${\displaystyle 0}$ and ${\displaystyle 1}$ are trivial perfect digit-to-digit invariants.

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

All natural numbers ${\displaystyle n}$ are preperiodic points for ${\displaystyle F_{b))$, regardless of the base. This is because all natural numbers of base ${\displaystyle b}$ with ${\displaystyle k}$ digits satisfy ${\displaystyle b^{k-1}\leq n\leq (k){(b-1)}^{b-1))$. However, when ${\displaystyle k\geq b+1}$, then ${\displaystyle b^{k-1}>(k){(b-1)}^{b-1))$, so any ${\displaystyle n}$ will satisfy ${\displaystyle n>F_{b}(n)}$ until ${\displaystyle n. There are a finite number of natural numbers less than ${\displaystyle b^{b+1))$, so the number is guaranteed to reach a periodic point or a fixed point less than ${\displaystyle b^{b+1))$, 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 ${\displaystyle b}$.

The number of iterations ${\displaystyle i}$ needed for ${\displaystyle F_{b}^{i}(n)}$ to reach a fixed point is the ${\displaystyle b}$-factorion function's persistence of ${\displaystyle n}$, 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 ${\displaystyle b}$.

Convention 00 = 1

Base Nontrivial perfect digit-to-digit invariants (${\displaystyle n\neq 1}$) Cycles
2 10 ${\displaystyle \varnothing }$
3 12, 22 2 → 11 → 2
4 131, 313 2 → 10 → 2
5 ${\displaystyle \varnothing }$

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

Convention 00 = 0

Base Nontrivial perfect digit-to-digit invariants (${\displaystyle n\neq 0}$, ${\displaystyle n\neq 1}$)[1] Cycles
2 ${\displaystyle \varnothing }$ ${\displaystyle \varnothing }$
3 12, 22 2 → 11 → 2
4 130, 131, 313 ${\displaystyle \varnothing }$
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 ${\displaystyle \varnothing }$ ${\displaystyle \varnothing }$
12 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 ${\displaystyle 0^{0}=1}$.

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

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

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