ALychrel number is anatural number that cannot form apalindrome through theiterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the196-algorithm, after the most famous number associated with the process. Inbase ten, no Lychrel numbers have been yetproven to exist, but many, including 196, are suspected onheuristic^{[1][better source needed]} andstatistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a roughanagram of "Cheryl", his girlfriend's first name.^[2]

The reverse-and-add process produces the sum of a number and the number formed by reversing the order of its digits. For example, 56 + 65 = 121. As another example, 125 + 521 = 646.

Some numbers become palindromes quickly after repeated reversal and addition, and are therefore not Lychrel numbers. All one-digit and two-digit numbers eventually become palindromes after repeated reversal and addition.

About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers:

• 56 becomes palindromic after one iteration: 56+65 =121.
• 57 becomes palindromic after two iterations: 57+75 = 132, 132+231 =363.
• 59 becomes a palindrome after three iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111
• 89 takes an unusually large24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome8813200023188.
• 10,911 reaches the palindrome4668731596684224866951378664 (28 digits) after55 steps.
• 1,186,060,307,891,929,990 takes261 iterations to reach the 119-digit palindrome44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which was a former world record for theMost Delayed Palindromic Number. It was solved by Jason Doucette's algorithm and program (usingBenjamin Despres' reversal-addition code) on November 30, 2005.
• On January 23, 2017 a Russian schoolboy, Andrey S. Shchebetov, announced on his website that he had found a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. This sequence was published in theOEIS asA281506. This sequence started with 1,186,060,307,891,929,990 – by then the only publicly known number found by Jason Doucette back in 2005. On May 12, 2017 this sequence was extended to 108864 terms in total and included the first 108864 delayed palindromes with 261-step delay. The extended sequence ended with 1,999,291,987,030,606,810 – its largest and its final term.
• On 26 April 2019, Rob van Nobelen computed a new world record for the Most Delayed Palindromic Number: 12,000,700,000,025,339,936,491 takes288 iterations to reach a 142 digit palindrome6634343445544188178365154497662249922269477578658488045222897505659677887769565057982225408848568757749622299422667944515638718814455443434366.
• On 5 January 2021, Anton Stefanov computed two new Most Delayed Palindromic Numbers:13968441660506503386020 and13568441660506503386420 take 289 iterations to reach the same 142 digit palindrome as the Rob van Nobelen number.
• On December 14, 2021, Dmitry Maslov computed a new world record for the Most Delayed Palindromic Number:1,000,206,827,388,999,999,095,750 takes293 iterations to reach 132 digit palindrome880226615529888473330265269768646444333433887733883465996765424854458424567699564388337788334333444646867962562033374888925516622088
• The OEIS sequenceA326414 contains 19353600 terms with 288-step delay known at present.
• Any number fromA281506 could be used as a primary base to construct higher order 261-step palindromes. For example, based on 1,999,291,987,030,606,810 the following number 199929198703060681000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001999291987030606810 also becomes a 238-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 after 261 steps.

The smallest number that is not known to form a palindrome is196. It is therefore the smallest Lychrel number candidate.

The number resulting from the reversal of the digits of a Lychrel number not ending in zero is also a Lychrel number.

Let${\displaystyle n}$ be a natural number. We define theLychrel function for anumber baseb > 1,${\displaystyle F_b:\mathbb{N}\to \mathbb{N}}$, to be the following:

${\displaystyle F_b(n)=n+\sum _{i=0}^{k-1}{d}_i{b}^{k-i-1}}$

where${\displaystyle k=\lfloor {\log }_{b}n\rfloor +1}$ is the number of digits in the number in base${\displaystyle b}$, and

${\displaystyle d_i=\frac{nmod{b}^{i+1}-n{modb}^i}{b^i}}$

is the value of each digit of the number. A number is aLychrel number if there does not exist a natural number${\displaystyle i}$ such that${\displaystyle F_b^{i+1}(n)=2F_b^i(n)}$, where${\displaystyle F^i}$ is the${\displaystyle i}$-thiteration of${\displaystyle F}$

In otherbases (these bases arepowers of 2, likebinary andhexadecimal), certain numbers can be proven to never form a palindrome after repeated reversal and addition,^[3] but no such proof has been found for 196 and other base 10 numbers.

It isconjectured 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 absolutely demonstrated to be non-Lychrel are informally called "candidate Lychrel" numbers. The first few candidate Lychrel numbers (sequenceA023108 in theOEIS) 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.^[4] Wade Van Landingham's site lists the total number of found suspected Lychrel seed numbers for each digit length.^[5]

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.^[6] However, so far no algorithm has been developed to circumvent the reversal and addition iterative process.

The termthread, coined by Jason Doucette, refers to thesequence of numbers that may or may not lead to a palindrome through the reverse and add process. Any givenseed and its associatedkin numbers will converge on the same thread. The thread does not include the originalseed orkin number, but only the numbers that are common to both, after they converge.

Seed numbers are asubset 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 byKoji Yamashita in 1997.

Because196 (base-10) is the smallest candidate Lychrel number, it has received the most attention.

In the 1980s, the 196 palindrome problem attracted the attention ofmicrocomputer hobbyists, with search programs byJim Butterfield and others appearing in several mass-market computing magazines.^[7][8][9] In 1985 a program by James Killman ran unsuccessfully for over 28 days, cycling through 12,954 passes and reaching a 5366-digit number.^[9]

John Walker began his 196 Palindrome Quest on 12 August 1987 on aSun 3/260 workstation. He wrote aC program to perform the reversal and addition iterations and to check for a palindrome after each step. The program ran in thebackground with a low priority and produced a checkpoint to a file every two hours and when the system was shut down, recording the number reached so far and the number of iterations. It restarted itself automatically from the last checkpoint after every shutdown. It ran for almost three years, then terminated (as instructed) on 24 May 1990 with the message:

Stop point reached on pass 2,415,836.

Number contains 1,000,000 digits.

The sequence starting with 196 had grown to a number of one million digits after 2,415,836 iterations without reaching a palindrome. Walker published his findings on the internet along with the last checkpoint, inviting others to resume the quest using the number reached so far.

In 1995, Tim Irvin and Larry Simkins used amultiprocessor computer and reached the two million digit mark in only three months without finding a palindrome. Jason Doucette then followed suit and reached 12.5 million digits in May 2000. Wade VanLandingham used Jason Doucette's program to reach 13 million digits, a record published in Yes Mag: Canada's Science Magazine for Kids. Since June 2000, Wade VanLandingham has been carrying the flag using programs written by various enthusiasts. By 1 May 2006, VanLandingham had reached the 300 million digit mark (at a rate of one million digits every 5 to 7 days). Usingdistributed processing,^[10] 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.^[11] 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.^[12]

Inbase 2, 10110 (22 in decimal) has been proven to be a Lychrel number, since after 4 steps it reaches 10110100, after 8 steps it reaches 1011101000, after 12 steps it reaches 101111010000, and in general after 4n steps it reaches a number consisting of 10, followed byn + 1 ones, followed by 01, followed byn + 1 zeros. This number obviously cannot be a palindrome, and none of the other numbers in the sequence are palindromes.

Lychrel numbers have been proven to exist in the following bases: 11, 17, 20, 26, and all powers of 2.^[13][14][15]

No base contains any Lychrel numbers smaller than the base. In fact, in any given baseb, no single-digit number takes more than two iterations to form a palindrome. Forb > 4, ifk <b/2 thenk becomes palindromic after one iteration:k +k = 2k, which is single-digit in baseb (and thus a palindrome). Ifk >b/2,k becomes palindromic after two iterations.

The smallest number in each base which could possibly be a Lychrel number are (sequenceA060382 in theOEIS):

Lychrel numbers can be extended to the negativeintegers by use of asigned-digit representation to represent each integer.

• Arithmetic dynamics
• Palindromic number

• OEISsequence A023108 (Positive integers which apparently never result in a palindrome under ...)
• John Walker – Three years of computing
• Tim Irvin – About two months of computing
• Jason Doucette – World records – 196 Palindrome Quest, Most Delayed Palindromic Number
• Benjamin Despres
• 196 and Other Lychrel Numbers by Wade VanLandingham
• Weisstein, Eric W."196-Algorithm".MathWorld.
• MathPages – Digit Reversal Sums Leading to Palindromes
• NumberPhile
• All known delayed palindromic numbers – Dmitry Maslov, MDPN project
• Testing a delayed palindrome – Dmitry Maslov, MDPN project

• Arithmetic dynamics
• Base-dependent integer sequences
• Unsolved problems in number theory

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