Lychrel number
LychrelNumber
2002-08-18 15:16:43
2007-04-21 21:33:07
Definition
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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}
\item $983+389 = 1372$
\item $1372+2731 = 4103$
\item $4103+3014 = 7117$
\end{itemize}
So in 3 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 (numbers colloquially called ``Lychrel numbers'' in base 10 are in fact just \emph{Lychrel candidates}). However, in base 2 for example, there have been numbers proven to be Lychrel numbers\footnote{\cite{walker} informs us that Ronald Sprague has proved that the number 10110, for example, is a Lychrel number is base 2.}. The first Lychrel candidate is 196:
\begin{itemize}
\item $196+691 = 887$
\item $887+788 = 1675$
\item $1675+5761 = 7436$
\item $7436+6347 = 13783$
\item $13783+38731 = 52514$
\item $52514+41525 = 94039$
\item $94039+93049 = 187088$
\item $187088+880781 = 1067869$
\item $\ldots$
\end{itemize}
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:
\begin{center}
\begin{tabular}{cc}
Range & Possible Lychrels \\
\hline
0 - 100 & 0 \\
100 - 1,000 & 2 \\
1,000 - 10,000 & 3 \\
10,000 - 100,000 & 69 \\
100,000 - 1,000,000 & 99 \\
10,000,000 - 100,000,000 & 1728 \\
100,000,000 - 1,000,000,000 & 29,813 \\
\end{tabular}
\end{center}
\begin{thebibliography}{9}
\bibitem{vl} Wade VanLandingham, \PMlinkexternal{196 And Other Lychrel Numbers}{http://www.p196.org/}
\bibitem{walker} John Walker, \PMlinkexternal{Three Years of Computing}{http://www.fourmilab.ch/documents/threeyears/threeyears.html}
\end{thebibliography}