Perfect digit-to-digit invariant

A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen number[1]) is a natural number that is equal to the sum of its digits each raised to the power of itself. A number n is a PDDI if and only if:

where 0 ≤ di ≤ 9 and di ∈ ℤ.

An example is 3435, as . The process of raising a number to the power of itself is known as tetration.

The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009.[2] Because each digit is raised by itself, this evokes the story of Baron Munchausen raising himself up by his own ponytail.[3] Narcissistic numbers follow a similar rule, but in the case of the narcissistics the powers of the digits are fixed, 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]

0 and 1 are PDDIs in any base (using the convention that 00 = 0). Apart from 0 and 1 there are only two other PDDIs in the decimal system, 3435 and 438579088 (sequence A046253 in the OEIS). Note that the second of these is only a PDDI under the convention that 00 = 0, but this is standard usage in this area.[5][6] For a few months, the number 3435 was for this reason the favorite number of Matt Parker, a stand-up mathematician, but he soon "got bored" and switched his favorite number to other types of numbers, such as narcissistic numbers and perfect numbers. Parker vehemently disagrees with the notion that 438579088 is another Münchhausen number.[7]

An example of a PDDI in another base is the quaternary number 313, or 55 in denary, as .

Proof of finitude

There are finitely many PDDIs in any base. This can be proven as follows:

Let be a base. Every PDDI in base is equal to the sum of its digits each raised to a power equal to the digit. This sum is less than or equal to , where is the number of digits in , because is the largest possible digit in base . Thus,
The expression increases linearly with respect to , whereas the expression increases exponentially with respect to . So there is some such that
There are finitely many natural numbers with fewer than k digits, so there are finitely many natural numbers satisfying the first inequality. Thus, there are only finitely many PDDIs in base .

Tables of PDDIs

Without considering numbers containing a (non-leading) zero, the following is an exhaustive list of PDDIs for integer bases up to 12 (excluding 1, a PDDI in all bases):[1]

Base PDDIs (in that base) PDDIs (denary representation)
2 10 (requires 00 = 1) 2
3 12, 22 5, 8
4 131, 313 29, 55
6 22352, 23452 3164, 3416
7 13454 3665
9 31, 156262, 1656547 28, 96446, 923362
10 3435 3435
12 3A67A54832 20017650854

When the convention is used the following numbers are also PDDIs (as well as 0, in all bases):

Base PDDIs (in that base) PDDIs (denary representation)
4 130 28
5 103, 2024 28, 264
8 400, 401 256, 257
9 30, 1647063, 34664084 27, 917139, 16871323
10 438579088 438579088

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.
  7. ↑ Numberphile (2012-01-13), 3435 - Numberphile, retrieved 2017-12-04
  • Parker, Matt. "3435". Numberphile. Brady Haran.
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