Perfect digit-to-digit invariant explained

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,^[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

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 ^{k -1} _{i =0} d _{i} b ^{i} . We define the function F _{b} : N → N as F _{b(n)} = \sum ^{k -1} _{i =0} {d ^{d _{i}} _{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 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 ^{k(n)} _{b} = 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 ^{i} _{b} (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 F_b for specific b

All numbers are represented in base

b .

Convention 0⁰ = 1

Base	Nontrivial perfect digit-to-digit invariants (n ≠ 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

Convention 0⁰ = 0

Base	Nontrivial perfect digit-to-digit invariants (n ≠ 0 , n ≠ 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

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 = nums = 0.0while 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

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

• Web site: Parker. Matt. 3435. Numberphile. Brady Haran. 2013-04-01. https://web.archive.org/web/20170413072736/http://www.numberphile.com/videos/3435.html. 2017-04-13. dead.

Notes and References

1. van Berkel. Daan. On a curious property of 3435. 2009. 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. Book: Parker. Matt. Things to Make and Do in the Fourth Dimension. 2014. Penguin UK. 9781846147654. 28. 2 May 2015.
5. http://www.magic-squares.net/narciss.htm Narcisstic Number
6. Book: Wells , David . The Penguin Dictionary of Curious and Interesting Numbers . Penguin . London . 1997 . 185 . 0-14-026149-4.

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