I've been googling around a little lately and have stumbled across the so called Lychrel problem. For a natural number $x$, let $Rx$ denote the number obtained by reversing the base-$10$ digits of $x$ (i.e. $R90 = 9$, $R345 = 543$, etc.). Then the question is whether, given some initial $x$, the sequence defined by $$x_{n + 1} = x_n + Rx_n \quad \quad \quad x_0 = x$$ eventually produces a palindrome (i.e. $Rx_n = x_n$ for some $n$). An initial value for which no palindrome is ever obtained is called a Lychrel number. It is an open question whether any Lychrel numbers exist at all. The smallest suspected Lychrel number is $x = 196$. I've been trying to find out whether anyone has ever done any serious mathematical work on the issue, but all I have been able to find are either computational efforts or trivial facts. Does anyone know of any serious publications about this question?

Thanks in advance.

  • $\begingroup$ i hope this one help a bit users.tmok.com/~pla/Lychrel/Lychrel.shtml $\endgroup$ – dato datuashvili Dec 28 '11 at 9:38
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    $\begingroup$ @dato Thanks, but these are pretty much what I wanted to exclude under "computational efforts" as the pages linked to on that page basically just list the properties of "small" numbers under various Lychrel related operations. Of course this data is useful in forming conjectures, but in this question I looking more for proven results. $\endgroup$ – user12014 Dec 28 '11 at 9:43
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    $\begingroup$ i see good lucks @PZZ $\endgroup$ – dato datuashvili Dec 28 '11 at 9:45
  • $\begingroup$ Have you looked at Richard Guy's book, Unsolved Problems In Number Theory, 3rd edition? $\endgroup$ – Gerry Myerson Dec 28 '11 at 14:33
  • $\begingroup$ @GerryMyerson I looked at the table of contents and couldn't find anything. $\endgroup$ – user12014 Dec 28 '11 at 21:58

There doesn't seem to be much ... But here are two interesting things i found in a quick search:

On Palindromes and Palindromic Primes
Hyman Gabai and Daniel Coogan
Mathematics Magazine
Vol. 42, No. 5 (Nov., 1969), pp. 252-254 
(article consists of 3 pages)
Published by: Mathematical Association of 

Stable URL: http://www.jstor.org/stable/2688705


Numerical palindromes and the 196 Problem: http://www.osaka-ue.ac.jp/zemi/nishiyama/math2010/196.pdf

  • $\begingroup$ Thanks. However, these are not really the things I had in mind. About Lychrel numbers, the first article just mentions the well known fact that they exist in base 2. The second article is just a way of showing that the only candidates below 1000 are 196 and 879 and a vague density intuition about the rarity of the palindromes... It seems maybe there really isn't much out there... $\endgroup$ – user12014 Dec 28 '11 at 9:54
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    $\begingroup$ i searched a number of math journals looking for this stuff in addition to scholar.google.com i've put an alert out on a few subscriptions that i have, if get anything i'll post it for ya $\endgroup$ – Ahmed Masud Dec 28 '11 at 10:07

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