(05-04-2021 03:29 AM)Albert Chan Wrote: There is also a mod-4 bucket, for even base n:

n = 4k: d_{4} ≡ d_{8} ≡ ... ≡ d_{4k} ≡ 0 (mod 4) → d_{2} ≡ d_{6} ≡ ... ≡ d_{4k-2} ≡ 2 (mod 4)

This one I have covered with my GCD partitioning, because the GCD between 4k and { 4, 8, ... 4k-4 } is obviously always 4 or some multiple of it. However, this:

(05-04-2021 03:29 AM)Albert Chan Wrote: n = 4k+2: d_{4} ≡ d_{8} ≡ ... ≡ d_{4k} ≡ 2 (mod 4) → d_{2} ≡ d_{6} ≡ ... ≡ d_{4k+2} ≡ 0 (mod 4)

is new to me. Looking back, you already mentioned it back in your base-10-by-hand post, and I think I finally understand it now: divisibility by 4 in 4k+2 bases takes the last two digits, and the constraint that the first of these has to be odd (thanks to divisibility by 2 partitioning) leads to this result.

This is an interesting improvement for 4k+2 bases, but I don't think it's generalizable further in a worthwhile manner, so 4k bases get no additional help.

(05-04-2021 03:29 AM)Albert Chan Wrote: With this, I confirmed there is no solution for 16 ≤ n ≤ 40

Conjecture: there are no solutions for N>14. No clue how to go about proving it though.