Perfect digit-to-digit invariant

Short description: Munchausen number

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

Definition

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

[math]\displaystyle{ F_b(1) = 1 }[/math] for all [math]\displaystyle{ b }[/math], 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 [math]\displaystyle{ 0^0 = 0 }[/math], both [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1 }[/math] are trivial perfect digit-to-digit invariants.

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

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

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

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

All numbers are represented in base [math]\displaystyle{ b }[/math].

Convention 0⁰ = 1

Base	Nontrivial perfect digit-to-digit invariants ([math]\displaystyle{ n \neq 1 }[/math])	Cycles
2	10	[math]\displaystyle{ \varnothing }[/math]
3	12, 22	2 → 11 → 2
4	131, 313	2 → 10 → 2
5	[math]\displaystyle{ \varnothing }[/math]	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 ([math]\displaystyle{ n \neq 0 }[/math], [math]\displaystyle{ n \neq 1 }[/math])	Cycles
2	[math]\displaystyle{ \varnothing }[/math]	[math]\displaystyle{ \varnothing }[/math]
3	12, 22	2 → 11 → 2
4	130, 131, 313	[math]\displaystyle{ \varnothing }[/math]
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	[math]\displaystyle{ \varnothing }[/math]	[math]\displaystyle{ \varnothing }[/math]
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 [math]\displaystyle{ 0^0 = 1 }[/math].

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 implements 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

See also

• Arithmetic dynamics
• Dudeney number
• Factorion
• Happy number
• Kaprekar's constant
• Kaprekar number
• Meertens number
• Narcissistic number
• Perfect digital invariant
• Sum-product number

External links

• Parker, Matt. "3435". Numberphile. Brady Haran. http://www.numberphile.com/videos/3435.html.