Folklore tells us that there is no such thing as an uninteresting integer. For, would such uninteresting integers exist, there would certainly be a smallest among them. But being the smallest uninteresting integer is a very interesting property, hence we should not have found it. The only way around this conundrum is to conclude that every integer is interesting.
Well, today three thousand four hundred thirty five got a little more interesting. Besides having a nice decimal representation, this number also possesses the following curious property:
3^3 + 4^4 + 3^3 + 5^5 = 3435
\]
The only numbers who have this property are 1 and 3435. I know this for a fact because an exhaustive computer search until \(10^{10}\) did not reveal any other then numbers mentioned above.
Furthermore, if \(n > 10^{10}\) then \(n > 9^9\log(n)\) for
\frac{10^{10}}{\log(10^{10})} = \frac{10^{10}}{10} = 10^{9} > 9^{9}
\]
Now, define \(\theta(n) := \sum c_{i}^{c_{i}}\) where the \(c_{i}\) are the numerals that express \(n\) in base 10. Then it can be shown that \(9^{9}\log(n) > \theta(n)\).
Now we know that 3435 is the only nontrivial number with such a special property.
