Definition:Perfect Digit-to-Digit Invariant

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Definition

A perfect digit-to-digit invariant is a number $n$ which is equal to the sum of the digits of $n$ each raised to the power of itself.


Sequence

The only known perfect digit-to-digit invariants are:

$0, 1, 3435, 438 \, 579 \, 088$

where, in this case, $0^0$ is taken to be $0$.


Examples

$1$ is a Perfect Digit-to-Digit Invariant

$1$ is a perfect digit-to-digit invariant:

$1 = 1^1$


$3435$ is a Perfect Digit-to-Digit Invariant

$3435$ is a perfect digit-to-digit invariant:

$3435 = 3^3 + 4^4 + 3^3 + 5^5$


$438 \, 579 \, 088$ is a Perfect Digit-to-Digit Invariant

$438 \, 579 \, 088$ is a perfect digit-to-digit invariant:

$438 \, 579 \, 088 = 4^4 + 3^3 + 8^8 + 5^5 + 7^7 + 9^9 + 0^0 + 8^8 + 8^8$


Also defined as

Some sources disallow the zero from use in the definition of a perfect digit-to-digit invariant.

This is because $0^0$ is more usually defined defined as $0^0 = 1$ rather than as $0^0 = 0$.

If we define $0^0 = 1$ then not only does $438 \, 579 \, 088$ no longer qualify as a perfect digit-to-digit invariant, then nor does $0$ itself.


Also known as

A perfect digit-to-digit invariant is also known as a Münchhausen number.


This comes from the idea that these perfect digit-to-digit invariants "raise themselves" similarly to how Baron Hieronymus von Münchhausen raised himself by riding a cannonball in the $1943$ film Münchhausen.

Some sources render the name as Munchausen, with or without the umlaut.


Sources

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