Perfect digit-to-digit invariant
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
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
See also
References
- van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038 [math.HO].
- 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.
- Daan van Berkel, On a curious property of 3435.
- Parker, Matt (2014). Things to Make and Do in the Fourth Dimension. Penguin UK. p. 28. ISBN 9781846147654. Retrieved 2 May 2015.
- Narcisstic Number, Harvey Heinz
- Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin. p. 185. ISBN 0-14-026149-4.
External links
- Parker, Matt. "3435". Numberphile. Brady Haran. Archived from the original on 2017-04-13. Retrieved 2013-04-01.