Every number is interesting http://en.wikipedia.org/wiki/Interesting_number_paradox

Or no number is interesting. The argument never proves the base case. ;)

If there was a number of interesting numbers, that number should include itself in the count, since we clearly have some interest in this topic. So if the number of interesting numbers is 0, then zero is an interesting number hence the number of interesting numbers is at least 1.

Alternately: if there are no interesting numbers, that fact is itself interesting. But that fact can be encoded as a number (see Gödel), and the number must be interesting if the fact is interesting. Thus there is at least one interesting number.

The fact being interesting doesn't mean its encoding is interesting. I might find a movie interesting but looking at the raw bytes of the MP4 file much less so ;)

Does such a thing actually exist outside of the encoding used to represent it?

Uh oh, I think we may have run into... PHILOSOPHY!

http://xkcd.com/903/

Surely there are only 95 repeated pairs of three digits between 100100 and 194194? And another 95 symmetric numbers. With a leading zero you have 195 each way.