Editing Lychrel number (section)

== Reverse-and-add process ==
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 large [http://www.jasondoucette.com/pal/89 24 iterations] (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome ''8813200023188''.
*10,911 reaches the palindrome ''4668731596684224866951378664'' (28 digits) after [http://www.jasondoucette.com/pal/10911 55 steps].
*1,186,060,307,891,929,990 takes '''[http://www.jasondoucette.com/pal/1186060307891929990 261 iterations]''' to reach the 119-digit palindrome ''44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544'', which was a former world record for the [http://www.jasondoucette.com/worldrecords.html#Most Most Delayed Palindromic Number]. It was solved by Jason Doucette's algorithm and program (using [[Benjamin 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 the [[OEIS]] as [[oeis:A281506|A281506]]. 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 takes '''[http://jasondoucette.com/pal/12000700000025339936491 288 iterations]''' to reach a 142 digit palindrome ''6634343445544188178365154497662249922269477578658488045222897505659677887769565057982225408848568757749622299422667944515638718814455443434366''. 
*On 5 January 2021, Anton Stefanov computed two new Most Delayed Palindromic Numbers: [http://jasondoucette.com/pal/13968441660506503386020 13968441660506503386020] and [http://jasondoucette.com/pal/13568441660506503386420 13568441660506503386420] 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: [https://dmaslov.me/mdpn/checkpal.html?n=1000206827388999999095750 1,000,206,827,388,999,999,095,750] takes '''293 iterations''' to reach 132 digit palindrome ''880226615529888473330265269768646444333433887733883465996765424854458424567699564388337788334333444646867962562033374888925516622088''
*The OEIS sequence [[oeis:A326414|A326414]] contains 19353600 terms with 288-step delay known at present.
*Any number from [[oeis:A281506|A281506]] 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 is [[196 (number)|196]]. 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.

=== Formal definition of the process ===
Let <math>n</math> be a natural number. We define the '''Lychrel function''' for a [[number base]] ''b''&nbsp;>&thinsp;1, <math>F_b : \mathbb{N} \rightarrow \mathbb{N}</math>, to be the following:
:<math>F_b(n) = n + \sum_{i=0}^{k-1} d_i b^{k - i - 1}</math>
where <math>k = \lfloor \log_{b} n \rfloor + 1</math> is the number of digits in the number in base <math>b</math>, and 
:<math>d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i}</math>
is the value of each digit of the number. A number is a '''Lychrel number''' if there does not exist a natural number <math>i</math> such that <math>F_b^{i+1}(n) = 2 F_b^i(n)</math>, where <math>F^i</math> is the <math>i</math>-th [[Iterated function|iteration]] of <math>F</math>