Definition:Lychrel Number
Definition
A Lychrel number is a natural number which cannot form a palindromic number through repeated iteration of the reverse-and add process.
Candidate Lychrel Numbers
No natural number has been proved to be a Lychrel number as of time of writing (June $2017$). However, plenty of numbers have not shown themselves to terminate in a palindromic number, although in some cases millions of iterations have been tested.
Hence a candidate Lychrel number is a natural number which is not known to form a palindromic number through repeated iteration of the reverse-and add process.
Also defined as
Some sources use the term Lychrel number to refer to numbers whose Lychrel status is unknown, that is, to candidate Lychrel numbers.
Others, referred to in some circles as purists, reserve the term Lychrel number for a candidate Lychrel number which cannot be obtained from a previous candidate Lychrel number during the course of a sequence of reverse-and add operations.
Such candidate Lychrel numbers can also be referred to as seed or root numbers Lychrel.
The candidate Lychrel numbers which can be derived from others can sometimes be found referred to as kin numbers.
Also see
Historical Note
David Wells, reporting on the state of play in $1986$ in his Curious and Interesting Numbers, noted that P.C. Leyland had performed $50 \, 000$ reversals, which produced a number of over $26 \, 000$ digits, still not palindromic.
The same source reported that P. Anderson had continued the search to $70 \, 928$ digits without any success at reaching a palindrome.
By $1997$, in his Curious and Interesting Numbers, 2nd ed., he was able to report that John Walker and Tim Irvin had carried the process for $196$ to over one million digits, which was reached after $2 \, 415 \, 836$ reverse-and-add iterations, which took $3$ years of spare time on a Sun $3/260$.
They subsequently continued the work, taking the calculation to $2 \, 000 \, 000$ digits on a supercomputer, taking a mere $2$ months.
Still no sign of termination.
The term Lychrel number was coined in $2002$ by Wade VanLandingham, as a near-anagram of the name of his girlfriend Cheryl.
It's anyone's guess how to pronounce it.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $196$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $196$
- Weisstein, Eric W. "Lychrel Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LychrelNumber.html