BrowserUk,
Yes, I noticed a pattern right away but no way to get to it. Starting from the fact that regardless of what base the number is written in, they must all follow: 
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111

[download]
Essentially what you get is the powerset of the sum of base X from 0 to the left most significant digit. Unfortunately, I can't think of way to benefit from this without memoization.

Cheers - L~R

In reply to Re^2: Can It Be Written In Base X With Only 1s And 0s by Limbic~Region
in thread Can It Be Written In Base X With Only 1s And 0s by Limbic~Region

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