@Conflux posted a market about whether they would find a "nice number" above 69. This market gives a high-level sketch of a particular algorithm for potentially finding new nice numbers.

If the above market resolves "YES", then I will code the algorithm in Python and run it on my 2021 MacBook Pro. If the program finds any (new) nice number within 3 hours then this resolves YES, otherwise NO. If the linked market isn't resolved, or if we don't know the base of the number, this resolves N/A.

The algorithm will work like this: I will consider only solutions where both the square and the cube can be described in the base b in terms of concatenations of less than 10 sequences of base-b-digits where each sequence is an arithmetic progression (of numbers less than b). I will then create closed form formulas for both the square and cube based on the length of these sequences, using the closed-form formula for (sum for x from i to j of b^x (x + c)). I will then set the cube of the square equal to the square of the cube to get a formula in terms of the lengths and b which, if satisfied, indicates the existence of a nice number (note that it will also have to be the case that the lengths of the sequences that make up the square and the cube must sum to the number of digits in the square and the cube).

I will then search for length values that satisfy these constraints. This may end up being a brute-force search, but if I spot any obvious optimizations, I'll implement those too/If someone else writes the program and includes the optimizations, I'll accept that.

Close date will be 2023-03-31 3:59 pm.