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 ============================================================================== 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 ==============================================================================