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.
Although my above post illustrates an interesting property of 3435, the proof is not entirely correct. The proof can be salvaged, the inequalities change a bit and the upper bound get's twice as large.
ReplyDeleteI am working on an article which will give the proper proof.
1, 2, 1, 5, 8, 1, 29, 55, 1, 1, 3164, 3416, 1, 3665, 1, 1, 28, 96446, 923362, 1, 3435, ...
ReplyDeleteI submitted the above sequence to the online encyclopedia of integer sequences. It contains all "Munchausen" numbers, numbers which have similar properties as 3435 in different bases.
We start with base 2 and every 1 in the sequence starts a new base.
The online encyclopedia of integer sequences contains the above sequence under the identifier A166623
ReplyDeleteI asked Neil Sloane, the creator of the OEIS, to endorse me on arXiv. He kindly endorsed me so no I have uploaded my article about Munchausen numbers to the arXiv
ReplyDeleteWith a little delay and a few additions to the article the arXiv now contains my article: "On a curious property of 3435".
ReplyDeleteSee http://arxiv.org/abs/0911.3038