Definition:Perfect Digit-to-Digit Invariant
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
- Weisstein, Eric W. "Münchhausen Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MuenchhausenNumber.html
- Weisstein, Eric W. "Narcissistic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NarcissisticNumber.html