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 a power equal to the digit.

n=d_{k}^{{d_{k}}}+d_{{k-1}}^{{d_{{k-1}}}}+\dots +d_{2}^{{d_{2}}}+d_{1}^{{d_{1}}}\,.

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.^[2]^[3]

3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435

4^{4}+3^{3}+8^{8}+5^{5}+7^{7}+9^{9}+0^{0}+8^{8}+8^{8}

=256+27+16777216+3125+823543+387420489+0+16777216+16777216=438579088

More generally, 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

In all bases 1 is a PDDI.
In base 3 there are 2 PDDI's, namely 12 and 22. (5 and 8 in decimals)
In base 4 there are 2 PDDI's, namely 131 and 313. (29 and 55 in decimals)
In base 6 there are 2 PDDI's, namely 22352 and 23452. (3164 and 3416 in decimals)
In base 7 there is 1 PDDI's, namely 13454. (3665 in decimals)
In base 9 there are 3 PDDI's, namely 31, 156262 and 1656547. (28, 96446 and 923362 in decimals)

When the convention $0^{0}=0$ is used the following numbers are also PDDI's

In all bases 0 is a PDDI.
In base 4 there is one additional PDDI, namely 130. (28 in decimal)
In base 5 there are 2 PDDI's, namely 103 and 2024. (28 and 264 in decimal)
In base 8 there are 2 PDDI's, namely 400 and 401. (256 and 257 in decimal)
In base 9 there are 3 additional PDDI's, namely 30, 1647063 and 34664084. (27, 917139 and 16871323 in decimal)

References

↑ van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038 [math.HO].
↑ 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 – Matt's Ex". Numberphile. Brady Haran.

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