From: dittmer@OsFhRz70.Rz.Fh-Osnabrueck.De (Ingo Dittmer)
Newsgroups: sci.math
Subject: Re: Palindromic Numbers
Date: 1 Apr 1996 06:24:40 GMT
>dittmer@OsFhRz70.Rz.Fh-Osnabrueck.De (Ingo Dittmer) wrote:
>> In base 3 there are numbers which never reach a palindrome. As far
>> as I know, no such result has been proved for base 10. So, try again
>> with 196!
>Could you provide a reference for this base 3 result? I know it's
>easy to prove for bases 2,4,etc, but I didn't know it had been proved
>for any base except powers of 2.
My database says there is at least one number in base 3 which never
reaches a palindrom. But sorry: due to a hardware error some time ago
I have no reference for it. If my own memory is right, it was mentioned
in one of those "popular math" articles in one of those journals like
Scientific American (or the german Spektrum der Wissenschaft), Byte or
like that. Roughly in the years 1985--92. Since the base 3 result has been
proved there, the proof should not be too complicated. Or they left
out something, I don't know any more.
Ingo
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From: kfoster@rainbow.rmii.com (Kurt Foster)
Newsgroups: sci.math
Subject: Re: Palindromic Numbers
Date: 1 Apr 1996 14:41:30 GMT
Ingo Dittmer (dittmer@OsFhRz70.Rz.Fh-Osnabrueck.De) wrote:
: >dittmer@OsFhRz70.Rz.Fh-Osnabrueck.De (Ingo Dittmer) wrote:
: >> In base 3 there are numbers which never reach a palindrome. As far
: >> as I know, no such result has been proved for base 10. So, try again
: >> with 196!
: >Could you provide a reference for this base 3 result? I know it's
: >easy to prove for bases 2,4,etc, but I didn't know it had been proved
: >for any base except powers of 2.
: My database says there is at least one number in base 3 which never
: reaches a palindrom. But sorry: due to a hardware error some time ago
: I have no reference for it.
:
There was an article about the old palindrome conjecture in one of
Martin Gardner's "Mathematical Games" columns in Scientific American many
years ago. The column mentioned the "reverse the digits and add"
procedure had been carried out tens of thousands of times on 196
(decimal) without getting a palindromic number.
My recollection is that the column said the conjecture had not been
proved true in any base, and had been proved false only in the *binary*
or base two. It gave the first known counterexample, and the names of
the people who'd proved it to be such. The column also gives the outline
of the proof -- basically, that a certain nonpalindromic digit pattern
always occurs right in the middle of the sum every time.
==============================================================================
From: dseal@armltd.co.uk (David Seal)
Newsgroups: sci.math
Subject: Re: Palindromic Numbers
Date: 2 Apr 1996 14:22:16 GMT
kfoster@rainbow.rmii.com (Kurt Foster) writes:
> My recollection is that the column said the conjecture had not been
>proved true in any base, and had been proved false only in the *binary*
>or base two. It gave the first known counterexample, and the names of
>the people who'd proved it to be such. The column also gives the outline
>of the proof -- basically, that a certain nonpalindromic digit pattern
>always occurs right in the middle of the sum every time.
Here are some preliminary results from a search for counterexamples
that I've done - I first reported these when the same question came up
in rec.puzzles in January. I haven't managed to do anything more with
them since :-(, but hope to do something in the next few weeks.
***************************************************************************
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.
***************************************************************************
David Seal
dseal@armltd.co.uk
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