In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b {\displaystyle b} that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because 3435 = 3 3 + 4 4 + 3 3 + 5 5 {\displaystyle 3435=3^{3}+4^{4}+3^{3}+5^{5}} . The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009, as this evokes the story of Baron Munchausen raising himself up by his own ponytail because each digit is raised to the power of itself.
Definition Let n {\displaystyle n} be a natural number which can be written in base b {\displaystyle b} as the k-digit number d k − 1 d k − 2 . . . d 1 d 0 {\displaystyle d_{k-1}d_{k-2}...d_{1}d_{0}} where each digit d i {\displaystyle d_{i}} is between 0 {\displaystyle 0} and b − 1 {\displaystyle b-1} inclusive, and n = ∑ i = 0 k − 1 d i b i {\displaystyle n=\sum _{i=0}^{k-1}d_{i}b^{i}} . We define the function F b : N → N {\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} } as F b ( n ) = ∑ i = 0 k − 1 d i d i {\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}{d_{i}}^{d_{i}}} . (As 00 is usually undefined, there are typically two conventions used, one where it is taken to be equal to one, and another where it is taken to be equal to zero.) A natural number n {\displaystyle n} is defined to be a perfect digit-to-digit invariant in base b if F b ( n ) = n {\displaystyle F_{b}(n)=n} . For example, the number 3435 is a perfect digit-to-digit invariant in base 10 because 3 3 + 4 4 + 3 3 + 5 5 = 27 + 256 + 27 + 3125 = 3435 {\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435} .
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