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Lychrel number

From Wikipedia, the free encyclopedia

Question, Web Fundamentals.svg Unsolved problem in mathematics:
Do any base-10 Lychrel numbers exist?
(more unsolved problems in mathematics)

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 proved to exist, but many, including 196, are suspected on heuristic[1] and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend's first name.[2]

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Transcription

Hello everybody, welcome to this new episode of Micmaths ! Today I'm going to talk about math problems that are still open, meaning mathematical problems with no solution yet, even the greatest mathematicians haven't found the solution. When we talk about this kind of problem, the first thing that comes to mind, is that since the best minds of humanity didn't find the answer, the problem must be so complicated that common mortals can not even understand the question. Well sometimes it's wrong There are unresolved problems that a 10 year old kid can understand. Let me introduce you my favorite 5. Number 5 : Syracuse's conjecture. Take a random number. If this number is even, divide it by two. And if it is odd multiply it by 3 and add 1. If for example you chose number 13, 13 is an odd number, so you do 13 × 3 + 1 = 40. Then you repeat the process with the new number. 40 is even, so you divide by 2 and you get 20. Then you start again 20 ÷ 2 = 10. 10 ÷ 2 = 5 5 × 3 + 1 = 16 16 ÷ 2 = 8 8 → 4 → 2 → 1 → 4 → 1 → 2 → ... you see that starting 13, you end up trapped in cycle 4 → 2 → 1 that repeats endlessly. The question is: are there numbers of departure which do not end in this cycle 4 → 2 → 1? You can try with different starting numbers, it seems that every time we get, after a number of steps, this cycle 4 → 2 → 1. In fact, all the numbers which were tested, until now, by mathematicians result in this cycle 4 → 2 → 1. Is there one that we don't find yet and and which behaves differently? That, for the moment, we do not know. 4: the Ramsey's numbers. Take a certain number of points. For example 5, like this and connect all the points with a blue or red lines. Is it possible to find a group of points which are all connected by the same color? In the example, we can see that these three points are all connected in blue. If you erase all the other points, it remains only the blue lines. However, if we look the other case, it is impossible to find 3 points all connected by the same color. There are no three points connected by red or three points connected in blue. In other words, if you have 5 points initially, it is sometimes possible to find 3 points all connected by the same color, and sometimes, it's impossible. However if initially you had 6 points instead of 5, this time, it's for sure that you can always find three points connected by the same color. You can try it at home: on a sheet of paper, connect 6 points with blue or red lines, you will see that it is impossible to have neither blue or red triangle. But is there in the same way a threshold from which there are always 4 points connected by the same color? The answer is yes. This threshold was found by mathematicians, it is 18. From 18 points, all connected by blue or red lines, you will inevitably find 4 all connected by blue or red. And we can continue: is there a threshold from which there are always 5 points connected by the same color? And this is where the unsolved problem happens, because the threshold, we know it exists, but we don't know how much is it. Finally mathematicians have found an interval because we know that this threshold is between 43 and 49 points, but it is not known exactly how much it is. 3: the Lychrel's numbers. A palindrome is a number that reads the same way from left to right or right to left. For example, 272 is a palidrome because it is written 2-7-2 whatever the direction in which we read If we take a number that is not a palindrome and adds that its inverse that is to say, himself written backwards, it is common that we obtains a palindrome. If you take for example the number 143, and add its inverse 341, you get a palindrome: 484. And if it does not work the first time, it is possible to repeat this operation. If you take the number 57, adding its inverse 75, we obtains 132. 132 is not a palindrome, but if you add its inverse 231 363 is obtained which is a palindrome. But there are numbers that never become a palindrome, whatever the number of steps that you will do? A number like this is called a Lychrel's number, but we do not know if it exists. There are still numbers which we suspect of being Lychrel's numbers it's the case for example of 196. If we add its inverse 691, we obtains 887. If we add to 887 its inverse, 788 we obtains 1675, which is not a palindrome. And so on, we can repeat this with different numbers obtained, we nerver obtains a palindrome number. Anyway, as far as may have been the calculations of mathematicians, we have still not found palindromes starting from 196, but that does not mean we will never find it. 196 is a Lychrel's number suspected, but it's still not proven. 2: the chromatic number of the plane. Take a large sheet of paper and a stick. You want to fully coloring sheet with a number of colors, so if you put the stick on the sheet, both ends can not be in areas coloured by the same color. If you paint the sheet like this, it does not work because if you put the stick like this, both ends of the stick are in blue areas. You can find a coloring that works using 7 colors distributed in a hexagonal tiling like this. You can put the stick as you want, it is not possible that both ends are located in areas coloured by the same color. Is it possible to do the same, but with less than 7 colors? Well, we do not know the answer. As earlier for Ramsey's numbers, mathematicians have found an interval. The minimum number of colors needed to make this possible, called the chromatic number of the plane, is between 4 and 7. It is either 4, or 5, or 6, or 7. But is it possible with 6, 5 or 4, we do not know. 1: multiplicative persistence of numbers. Take a number and multiply its numbers between them. If for example you take 73, it's composed of a 7 and a 3, so you do 7 × 3 = 21. Then you repeat this with the obtained number. With 21 we obtains 2 × 1 = 2. And now you got a 1-digit number, you stop. From 73, it took 2 steps to reach a 1-digit number and stop. so we say that the multiplicative persistence of 73 is equal to 2. The question is: can we find numbers with multiplicative persistence as big as we want? It seems that the answer is "no" but at the moment, no one knows how to prove it. For now, the number with the biggest persistence that we have found is 277,777,788,888,899. This number's persistence is equal to 11. So, it takes 11 steps before getting a 1-digit number. But there are numbers with greater multiplicative persistence than 11? Nobody knows, at least for now, we have not found. Well, this video is over, I hope that you enjoyed. Perhaps she tempted you to start attack these problems! Maybe you are the the young mathematicians who will solve some of these problems ... So good research and soon for new mathematical discoveries :)

Contents

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 3 iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111
  • 89 takes an unusually large 24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8,813,200,023,188.
  • 10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after 55 steps.
  • 1,186,060,307,891,929,990 takes 261 iterations to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which is the current world record for the 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 web site 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 OEIS as 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 288 iterations to reach a 142 digit palindrome.
  • Any number from 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 known number that is not known to form a palindrome is 196. It is the smallest Lychrel number candidate.

The number resulting from the reversal of the digits of a Lychrel number is also a Lychrel number.

Proof not found

In other bases (these bases are power of 2, like binary and hexadecimal), 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 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 (sequence A023108 in the OEIS) 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.

Threads, seed and kin numbers

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.

196 palindrome quest

Because 196 (base-10) is the lowest 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.[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 a Sun 3/260 workstation. He wrote a C program to perform the reversal and addition iterations and to check for a palindrome after each step. The program ran in the background 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.

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 a multiprocessor 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). Using distributed 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 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]

Other bases

In base 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 by n+1 ones, followed by 01, followed by n+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]

The smallest number in each base which could possibly be a Lychrel number are (sequence A060382 in the OEIS):

b Smallest possible Lychrel number in base b
written in base b (base 10)
2 10110 (22)
3 10211 (103)
4 10202 (290)
5 10313 (708)
6 4555 (1079)
7 10513 (2656)
8 1775 (1021)
9 728 (593)
10 196 (196)
11 83A (1011)
12 179 (237)
13 12CA (2701)
14 1BB (361)
15 1EC (447)
16 19D (413)
17 B6G (3297)
18 1AF (519)
19 HI (341)
20 IJ (379)
21 1CI (711)
22 KL (461)
23 LM (505)
24 MN (551)
25 1FM (1022)
26 OP (649)
27 PQ (701)
28 QR (755)
29 RS (811)
30 ST (869)

References

  1. ^ O'Bryant, Kevin (26 December 2012). "Reply to Status of the 196 conjecture?". Math Overflow.
  2. ^ "FAQ".
  3. ^ Brown, Kevin. "Digit Reversal Sums Leading to Palindromes". MathPages.
  4. ^ VanLandingham, Wade. "Lychrel Records". p196.org.
  5. ^ VanLandingham, Wade. "Identified Seeds". p196.org.
  6. ^ "On Non-Brute Force Methods". Archived from the original on 2006-10-15.
  7. ^ "Bits and Pieces". The Transactor. Transactor Publishing. 4 (6): 16–23. 1984. Retrieved 26 December 2014.
  8. ^ Rupert, Dale (October 1984). "Commodares: Programming Challenges". Ahoy!. Ion International (10): 23, 97–98.
  9. ^ a b Rupert, Dale (June 1985). "Commodares: Programming Challenges". Ahoy!. Ion International (18): 81–84, 114.
  10. ^ Swierczewski, Lukasz; Dolbeau, Romain (June 23, 2014). The p196_mpi Implementation of the Reverse-And-Add Algorithm for the Palindrome Quest. International Supercomputing Conference. Leipzig, Germany.
  11. ^ Dolbeau, Romain. "The p196_mpi page". www.dolbeau.name.
  12. ^ "Lychrel Records". Archived from the original on December 5, 2003. Retrieved September 2, 2016. Cite uses deprecated parameter |dead-url= (help)
  13. ^ See comment section in OEISA060382
  14. ^ "Digit Reversal Sums Leading to Palindromes".
  15. ^ "Letter from David Seal".

External links

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