What's Special About 3435?

The number 3435 has the honour of being the only base 10 Munchausen number. So what is a Munchausen number? Wikipedia provides this definition:

A Munchausen number is a natural number in a given number base \(b\) that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because \(3435=3^3+4^4+3^3+5^5\). 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.

Of course, the number 1 qualifies as well but this is trivial and can be ignored. The only rival to 3435 comes in the form of 438579088 but runs into the problem of what the value of \(0^0\) is. If we take \(0^0=0\) then it does qualify because:$$438579088 = 4^4+3^3+8^8+5^5+7^7+9^9+0^0+8^8+8^8$$However, \(0^0\) is also commonly evaluated as 1 (see link) and so there is an ambiguity surrounding 438579088. If we take \(0^0=0\) then it does qualify as a Munchausen number but if we take \(0^0=1\), it doesn't. 3425 however, suffers from no such ambiguity.

Let's spell out 3435's unique property in large type:

\(3435=3^3+4^4+3^3+5^5\)

There are Munchausen numbers in other number bases as well but in this post I'm just focusing on base 10. Here is a permalink for identifying Munchausen numbers in base 10.