As you can see, the square and cube of 69, in base 10, contain all the digits with no repeats. In general, a 

, in some base, is a real, natural number whose square and cube, written in that base, contain all the digits in that base with no repeats. Only normal positive integer bases are considered for this market.  

So far, this is the only nice number I know of. I suspect that there are an infinite number, but they only start appearing around base 120 to 130; I checked up to around base 30.

http://tinyurl.com/confluxblog/post/is-69-unique

UPDATE 1/30/23: I wrote an update blog post! 

http://tinyurl.com/confluxblog/post/progress-update-on-the-search-for-nice-numbers

Anyway, hopefully this market inspires you to do some research into finding nice numbers, or to find someone else online who has!

I may also provide mana rewards for partial results.

Note: The original version of this market, embedded below, had a poorly worded definition which included negative numbers such as -69 (and more arguably complex numbers such as 69i and -69i). I've cancelled that market (resolved it N/A) and made this replacement. Still, the old one contains many comments with progress on the problem and cool related results! Along with discussion over the N/A resolution. 

My new, subject-to-revision policy based on this dispute: In the rare event of a conflict between my resolution criteria and the common-sense, consensus spirit of the market, I may resolve it according to the market's spirit or N/A, probably after discussion. 