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 that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because . 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 be a natural number which can be written in base as the k-digit number where each digit is between and inclusive, and . We define the function as . (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 is defined to be a perfect digit-to-digit invariant in base b if . For example, the number 3435 is a perfect digit-to-digit invariant in base 10 because .

for all , 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 , both and are trivial perfect digit-to-digit invariants.

A natural number is a sociable digit-to-digit invariant if it is a periodic point for , where for a positive integer , and forms a cycle of period . A perfect digit-to-digit invariant is a sociable digit-to-digit invariant with . An amicable digit-to-digit invariant is a sociable digit-to-digit invariant with .

All natural numbers are preperiodic points for , regardless of the base. This is because all natural numbers of base with digits satisfy . However, when , then , 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. This means also that there are a finite number of perfect digit-to-digit invariant and cycles for any given base .

The number of iterations needed for to reach a fixed point is the -factorion function's persistence of , 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 .

Convention 00 = 1

Base Nontrivial perfect digit-to-digit invariants () Cycles
2 10
3 12, 22 2 → 11 → 2
4 131, 313 2 → 10 → 2
5

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 (, )[1] Cycles
2
3 12, 22 2 → 11 → 2
4 130, 131, 313
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
12 3A67A54832

Programming examples

Python

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

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

Java

The following program in Java determines whether an integer number is a Munchausen Number / Perfect Digit to Digit Invariant or not, following the convention .

import java.util.Scanner;
public class Munchausen
{
    public static void main ()
    {
        Scanner in = new Scanner (System.in);
       System.out.println("Enter number:");
       int num = in.nextInt(), temp = num, digit; double sum = 0;
       while (num>0)
       { digit = num % 10;
         num /= 10;
         sum += Math.pow(digit, digit);
        }
        
        if (sum == temp)
        System.out.print("Munchausen Number");
        else
        System.out.print("Not Munchausen Number");
    }
}

See also

References

  1. ^ a b van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038 [math.HO].
  2. ^ Olry, Regis and Duane E. Haines. "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.
  3. ^ Daan van Berkel, On a curious property of 3435.
  4. ^ Parker, Matt (2014). Things to Make and Do in the Fourth Dimension. Penguin UK. p. 28. ISBN 9781846147654. Retrieved 2 May 2015.
  5. ^ Narcisstic Number, Harvey Heinz
  6. ^ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin. p. 185. ISBN 0-14-026149-4.