Munchausen number

A Munchausen (or Münchhausen) number is a natural number n the sum of whose digits (in base 10), each raised to the power of itself, is n itself.^[1] In other words, if the number has the decimal representation

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

then

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

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 known 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

• Happy number
• Lucky number
• Perfect digit-to-digit invariant

External links

• Weisstein, Eric W., "Münchhausen Number", MathWorld.
• Numberphile – 3435

References

1. Strachan, Liz (2014). Numbers Are Forever. New York: Constable & Robinson. p. 70. ISBN 9781472111104. Retrieved 2 May 2015.
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. (sequence A046253 in OEIS)