Puzzle - RPL and others
05-04-2021, 06:48 AM
Post: #26
 3298 Member Posts: 206 Joined: Oct 2014
RE: Puzzle - RPL and others
(05-04-2021 03:29 AM)Albert Chan Wrote:  There is also a mod-4 bucket, for even base n:

n = 4k: ﻿ ﻿ ﻿ ﻿ d4 ≡ d8 ≡ ... ≡ d4k ≡ 0 (mod 4) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → d2 ≡ d6 ≡ ... ≡ d4k-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: d4 ≡ d8 ≡ ... ≡ d4k ≡ 2 (mod 4) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → d2 ≡ d6 ≡ ... ≡ d4k+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.
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