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:

n=\sum _{i=0}^{k}d_{i}10^{i}=d_{k}^{d_{k}}+d_{k-1}^{d_{k-1}}+\cdots +d_{1}^{d_{1}}+d_{0}^{d_{0}}

where 0 ≤ d_i ≤ 9 and d_i ∈ ℤ.

An example is 3435, as $3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435$ . 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 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 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] An example of a PDDI in another base is the quaternary number 313, or 29 in denary, as $3^{3}+1^{3}+3^{3}=123_{4}+1_{4}+123_{4}=313_{4}=29_{10}$ .

Proof of finitude

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

Let

b

be a base. Every PDDI

n

in base

b

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

a(b-1)^{{b-1}}

, where

a

is the number of digits in

n

, because

b-1

is the largest possible digit in base

b

. Thus,

a(b-1)^{{b-1}}\geq n\geq b^{{a-1}}.

The expression

a(b-1)^{{b-1}}

increases linearly with respect to

a

, whereas the expression

b^{{a-1}}

increases exponentially with respect to

a

. So there is some

k>0

such that

\forall a\geq k,\,\,a(b-1)^{{b-1}}<b^{{a-1}}.

There are finitely many natural numbers

n

with fewer than k digits, so there are finitely many natural numbers

n

satisfying the first inequality. Thus, there are only finitely many PDDIs in base

b

Tables of PDDIs

Without considering numbers containing a (non-leading) zero, the following is an exhaustive list of PDDIs for integer bases up to 10 (excluding 1, a PDDI in all bases):^[1]

Base	PDDIs (in that base)	PDDIs (denary representation)
2	10	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

When the convention $0^{0}=0$ 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 2 van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038  [math.HO].
↑ 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.
↑ Daan van Berkel, On a curious property of 3435.
↑ Parker, Matt (2014). Things to Make and Do in the Fourth Dimension. Penguin UK. p. 28. ISBN 9781846147654. Retrieved 2 May 2015.
↑ Narcisstic Number, Harvey Heinz
↑ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin. p. 185. ISBN 0-14-026149-4.

External links

Parker, Matt. "3435". Numberphile. Brady Haran.

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