| Version 4 |
Version 3 |
| \PMlinkescapeword{range} \PMlinkescapeword{ranges} |
\PMlinkescapeword{range} \PMlinkescapeword{ranges} |
| A \emph{Lychrel number} is a number which never yields a palindrome in the iterative process of adding to itself a copy of itself with digits reversed. For example, if we start with the number 983 we get: |
A \emph{Lychrel number} is a number which never yields a palindrome in the iterative process of adding to itself a copy of itself with digits reversed. For example, if we start with the number 983 we get: |
| \begin{itemize} |
\begin{itemize} |
| \item $983+389 = 1372$ |
\item $983+389 = 1372$ |
| \item $1372+2731 = 4103$ |
\item $1372+2731 = 4103$ |
| \item $4103+3104 = 7207$ |
\item $4103+3104 = 7207$ |
| \item $7207+7027 = 14234$ |
\item $7207+7027 = 14234$ |
| \item $14234+43241 = 57475$ |
\item $14234+43241 = 57475$ |
| \end{itemize} |
\end{itemize} |
| So in 5 steps we get a palindrome, hence 983 is not a Lychrel number. |
So in 5 steps we get a palindrome, hence 983 is not a Lychrel number. |
|
In fact, it is not known if there exist any Lychrel numbers (in base 10-- in base 2 for instance, there have been numbers proven to be Lychrel numbers\footnote{\cite{walker} informs us that Ronald Sprague has proved that the number 10110 in base 2 is a Lychrel number.}). The first Lychrel candidate is 196:
|
In fact, it is not known if there exist any Lychrel numbers (in base 10-- in base 2 for instance, there have been numbers proven to be Lychrel numbers). The first Lychrel candidate is 196:
|
| \begin{itemize} |
\begin{itemize} |
| \item $196+691 = 887$ |
\item $196+691 = 887$ |
| \item $887+788 = 1675$ |
\item $887+788 = 1675$ |
| \item $1675+5761 = 7436$ |
\item $1675+5761 = 7436$ |
| \item $7436+6347 = 13783$ |
\item $7436+6347 = 13783$ |
| \item $13783+38731 = 52514$ |
\item $13783+38731 = 52514$ |
| \item $52514+41525 = 94039$ |
\item $52514+41525 = 94039$ |
| \item $94039+93049 = 187088$ |
\item $94039+93049 = 187088$ |
| \item $187088+880781 = 1067869$ |
\item $187088+880781 = 1067869$ |
| \item $\ldots$ |
\item $\ldots$ |
| \end{itemize} |
\end{itemize} |
| This has been followed out to millions of digits, with no palindrome found in the sequence. |
This has been followed out to millions of digits, with no palindrome found in the sequence. |
| The following table gives the number of Lychrel candidates found within ascending ranges: |
The following table gives the number of Lychrel candidates found within ascending ranges: |
| \begin{center} |
\begin{center} |
| \begin{tabular}{cc} |
\begin{tabular}{cc} |
| Range & Possible Lychrels \\ |
Range & Possible Lychrels \\ |
| \hline |
\hline |
| 0 - 100 & 0 \\ |
0 - 100 & 0 \\ |
| 100 - 1,000 & 2 \\ |
100 - 1,000 & 2 \\ |
| 1,000 - 10,000 & 3 \\ |
1,000 - 10,000 & 3 \\ |
| 10,000 - 100,000 & 69 \\ |
10,000 - 100,000 & 69 \\ |
| 100,000 - 1,000,000 & 99 \\ |
100,000 - 1,000,000 & 99 \\ |
| 10,000,000 - 100,000,000 & 1728 \\ |
10,000,000 - 100,000,000 & 1728 \\ |
| 100,000,000 - 1,000,000,000 & 29,813 \\ |
100,000,000 - 1,000,000,000 & 29,813 \\ |
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
| \begin{thebibliography}{9} |
\section{References} |
| \bibitem{vl} Wade VanLandingham, \PMlinkexternal{196 And Other Lychrel Numbers}{http://www.p196.org/} |
\begin{itemize} |
| \bibitem{walker} John Walker, \PMlinkexternal{Three Years of Computing}{http://www.fourmilab.ch/documents/threeyears/threeyears.html} |
\item Wade VanLandingham, \PMlinkexternal{196 And Other Lychrel Numbers}{http://www.p196.org/} |
| \end{thebibliography} |
\item John Walker, \PMlinkexternal{Three Years of Computing}{http://www.fourmilab.ch/documents/threeyears/threeyears.html} |
|
\end{itemize} |
