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Munchausen number

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Title: Munchausen number  
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Subject: Baron Munchausen, Base-dependent integer sequences, Baron Munchausen's Dream, Rudolf Erich Raspe, The Adventures of Baron Munchausen
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Munchausen number

A Munchausen (or Münchhausen) number is a natural number n the sum of whose digits, each raised to the power of itself, is n itself.[1] In other words, if the number has the decimal representation

n = a_k a_{k-1} \cdots a_0,


n = \sum_{i=0}^k a_i^{a_i}.

The term was coined by Dutch mathematician and software engineer Daan van Berkel in 2009.[2] Because each digit is "raised up" by itself, this evokes the story of Baron Munchausen raising himself up by his own ponytail.[3] Narcissistic numbers follow a similar rule, but here the powers of the digits are fixed in some way, for example being raised to the power of the number of digits in the number. This is an additional explanation for the name, as Baron Münchhausen was famously narcissistic.[4]

One example is

3435=3^3+4^4+3^3+5^5 = 27+256+27+3125.

When discussing Munchausen numbers, the non-standard definition 00 = 0 is used,[5] yielding four Munchausen numbers in base 10:

  • 0,
  • 1,
  • 3435 and
  • 438,579,088.

Normally, however, 00 is considered to be undefined, a rule which would only allow 1 and 3435 to be accepted.

See also

External links


  1. ^ Strachan, Liz (2014). Numbers Are Forever. New York: Constable & Robinson. p. 70.  
  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.  
  5. ^ (sequence A046253 in OEIS)
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