Perfect digit-to-digit invariant

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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]

Contents

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

BaseNontrivial perfect digit-to-digit invariants ()Cycles
210
312, 222 → 11 → 2
4131, 3132 → 10 → 2
5

2 → 4 → 2011 → 12 → 10 → 2

104 → 2013 → 113 → 104

622352, 23452

4 → 1104 → 1111 → 4

23445 → 24552 → 50054 → 50044 → 24503 → 23445

71345412066 → 536031 → 265204 → 265623 → 551155 → 51310 → 12125 → 12066
8405 → 6466 → 421700 → 3110776 → 6354114 → 142222 → 421 → 405
931, 156262, 1656547
103435
11
123A67A54832

Convention 00 = 0

BaseNontrivial perfect digit-to-digit invariants (, ) [1] Cycles
2
312, 222 → 11 → 2
4130, 131, 313
5103, 2024

2 → 4 → 2011 → 11 → 2

9 → 2012 → 9

622352, 23452

5 → 22245 → 23413 → 1243 → 1200 → 5

53 → 22332 → 150 → 22250 → 22305 → 22344 → 2311 → 53

713454
8400, 401
930, 31, 156262, 1647063, 1656547, 34664084
103435, 438579088
11
123A67A54832

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=nums=0.0whilenum>0:digit=num%10num//=10s+=pow(digit,digit)ifs==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

defpddif(x:int,b:int)->int:total=0whilex>0:total=total+pow(x%b,x%b)x=x//breturntotaldefpddif_cycle(x:int,b:int)->list[int]:seen=[]whilexnotinseen:seen.append(x)x=pddif(x,b)cycle=[]whilexnotincycle:cycle.append(x)x=pddif(x,b)returncycle

Convention 00 = 0

defpddif(x:int,b:int)->int:total=0whilex>0:ifx%b>0:total=total+pow(x%b,x%b)x=x//breturntotaldefpddif_cycle(x:int,b:int)->list[int]:seen=[]whilexnotinseen:seen.append(x)x=pddif(x,b)cycle=[]whilexnotincycle:cycle.append(x)x=pddif(x,b)returncycle

See also

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References

  1. 1 2 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.