Личрел сандары және палиндромға айналмау процесі

A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196 algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proven to exist, but many, including 196, are suspected on heuristic and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of "Cheryl", his girlfriend's first name.

In other bases (these bases are powers of 2, like binary and hexadecimal), certain numbers can be proven to never form a palindrome after repeated reversal and addition, but no such proof has been found for 196 and other base 10 numbers. It is conjectured that 196 and other numbers that have not yet yielded a palindrome are Lychrel numbers, but no number in base ten has yet been proven to be Lychrel. Numbers which have not been demonstrated to be non Lychrel are informally called "candidate Lychrel" numbers. The first few candidate Lychrel numbers are:
196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997. The numbers in bold are suspected Lychrel seed numbers (see below). Computer programs by Jason Doucette, Ian Peters and Benjamin Despres have found other Lychrel candidates. Indeed, Benjamin Despres' program has identified all suspected Lychrel seed numbers of less than 17 digits. Wade Van Landingham's site lists the total number of found suspected Lychrel seed numbers for each digit length. The brute force method originally deployed by John Walker has been refined to take advantage of iteration behaviours. For example, Vaughn Suite devised a program that only saves the first and last few digits of each iteration, enabling testing of the digit patterns in millions of iterations to be performed without having to save each entire iteration to a file. However, so far no algorithm has been developed to circumvent the reversal and addition iterative process.

The term thread, coined by Jason Doucette, refers to the sequence of numbers that may or may not lead to a palindrome through the reverse and add process. Any given seed and its associated kin numbers will converge on the same thread. The thread does not include the original seed or kin number, but only the numbers that are common to both, after they converge. Seed numbers are a subset of Lychrel numbers, that is the smallest number of each non palindrome producing thread. A seed number may be a palindrome itself. The first three examples are shown in bold in the list above. Kin numbers are a subset of Lychrel numbers, that include all numbers of a thread, except the seed, or any number that will converge on a given thread after a single iteration. This term was introduced by Koji Yamashita in 1997.

Because 196 (base 10) is the smallest candidate Lychrel number, it has received the most attention. In the 1980s, the 196 palindrome problem attracted the attention of microcomputer hobbyists, with search programs by Jim Butterfield and others appearing in several mass market computing magazines. In 1985 a program by James Killman ran unsuccessfully for over 28 days, cycling through 12,954 passes and reaching a 5366 digit number. in 2011 Romain Dolbeau completed a billion iterations to produce a number with 413,930,770 digits, and in February 2015 his calculations reached a number with a billion digits. A palindrome has yet to be found. Other potential Lychrel numbers which have also been subjected to the same brute force method of repeated reversal addition include 879, 1997 and 7059: they have been taken to several million iterations with no palindrome being found.

Lychrel numbers can be extended to the negative integers by use of a signed digit representation to represent each integer.