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Originally found at: http://203.162.7.73/webs/math/mathpages/www.seanet.com/~ksbrown/dseal.htm

David Seal (dseal@armltd.co.uk) posted the following remarkable results in the Usenet Newsgroup rec.puzzles in January of 1996. I believe the results on powers of 2 are well known, but this is the first I've seen of results for other bases.

> OK, here is a quick summary of the results I've found about bases
> in which there are sequences which never go palindromic. In each
> case, I give a starting number for such a sequence and an indication
> of how it grows.
>
> The results were obtained by a search program, and should still be
> regarded as preliminary and unpolished. In particular, I've had to
> transcribe them from the program output by hand, and may have made
> errors in the process. I hope to complete the work sometime and
> produce a full description of the program's search method, properly
> verified results, etc.
>
> First, there is a regular family which can be shown to extend to any
> power of 2:
>
> Base 2:
> 10(n 1s)1101(n 0s)00
> After 4 iterations, becomes same thing with n increased by 1.
>
> Base 4:
> 10(n 3s)3323(n 0s)00
> After 6 iterations, becomes same thing with n increased by 1.
>
> Base 8:
> 10(n 7s)7767(n 0s)00
> After 8 iterations, becomes same thing with n increased by 1.
>
> Base 16:
> 10(n Fs)FFEF(n 0s)00
> After 10 iterations, becomes same thing with n increased by 1.
>
> Base 32:
> 10(n Vs)VVUV(n 0s)00
> After 12 iterations, becomes same thing with n increased by 1.
>
>
> Sporadic solutions:
>
> Base 4:
> 1033202000232(n 2s)2302333113230
> After 6 iterations, becomes same thing with n increased by 3.
>
> Base 11:
> 1246277(n As)A170352495681825A5026571A506181864A5143171(n 0s)0872542
> After 6 iterations, becomes same thing with n increased by 1.
>
> Base 17:
> 10023AB83E3B983CFGEC556G4G010(n 0s)0FGCG10FG505GF020CGF(n Gs)GG11G4F655D
> DGGB299B3D38BB320G
> After 6 iterations, becomes same thing with n increased by 1.
>
> Base 20:
> There is a >200 digit number of the same general form which grows
> indefinitely without ever producing a palindrome, but I'm not going
> to try to transcribe it here!
>
> Base 26:
> 1N5ELA6C(n Ps)P6E7(n 0s)0D59ME5N
> After 4 iterations, becomes same thing with n increased by 1.