Welcome to p196.org!
This site began as information on the number 196 only. Since then, it has kind of grown to include other Lychrel Numbers. Here are some of the other seeds that have been identified.
The first search that produced Lychrel Numbers was work done by Jason Doucette of Canada, and Ian Peters of England. Although neither of them were looking specifically for Lychrel Numbers, the work that they were doing led to the discovery of these numbers in an indirect way. I don't know if either one of them even created a list, but I do know that neither person ever published a list of the numbers they found.
Note: Jason has emailed me, questioning whether he should be given any credit for the above paragraph. Part of his note read like this: Although I did calculate which numbers did not solve out while I was trying to find the ones that took the longest to solve out, I did not compile or save this data. In other words - Although I did calculate specific Lychrel numbers (such as 196, 295, 887, etc), I did not calculate specific Lychrel threads (in which the copies are removed, such as 196, and NOT 295, then 887, etc) - since I did not save the data, I couldn't possibly have compiled it. In my opinion, he was still one of the first two people that I can see who did any work that would have identified Lychrel Numbers. A save function in his program would have compiled a list of numbers that he had "marked as infinite" (quoted from his site). Each person can decide for themselves, (Just as I don't agree that 9,999 qualifies as a Lychrel Number, but others might disagree.) but I still believe my above statement is true in spirit. End Note
Ian Peters states on his site that there are 1,895 Lychrel Numbers between 0-9,999,999. Jason doesn't give any reference to how many he found. On March 29, 2002 Ben Despres of the USA sent me a list of numbers between 0 and 99,999,999.
As a result, it is difficult for me to decide who should get credit as the discoverer of the numbers between 0 and 9,999,999. I have data that says Ian should be credited, but he has not responded to a few emails I have sent him asking for his list. I trust that he did discover the numbers, but I have no "proof". On the other hand, I have the lists that Ben sent........
I guess to be fair to my instincts, I am going to trust that Ian Peters discovered the first 1,895 Lychrel Numbers, and that they are the same ones Ben found. The dates that he discovered them is unclear to me. But in my mind, I am going to credit him with the discovery. If anyone disagrees, let me know, and I'll reconsider.
On the other hand, I can say with fair certainty, that Ben Despres deserves the credit for the discovery of all of the Seed Numbers between 10,000,000 and 99,999,999,999.
As far as how many of them are there, the following table shows the number of discovered Seed Numbers for a given number range:
|Number Range||Date Complete||Discoverer||App Coder||Seeds Found|
0 - 99
|Unknown||Ian Peters||Ian Peters||0|
100 - 999
|Unknown||Ian Peters||Ian Peters||2|
1,000 - 9,999
|Unknown||Ian Peters||Ian Peters||3|
10,000 - 99,999
|Unknown||Ian Peters||Ian Peters||69|
100,000 - 999,999
|Unknown||Ian Peters||Ian Peters||99|
1,000,000 - 9,999,999
|Unknown||Ian Peters||Ian Peters||1,728|
10,000,000 - 99,999,999
|May 29, 2002||Ben Despres||Ben Despres||1,651|
100,000,000 - 999,999,999
|June 25, 2002||Ben Despres||Ben Despres||28,162|
1,000,000,000 - 9,999,999,999
|January 5, 2003||Ben Despres||Ben Despres||25,780|
10,000,000,000 - 99,999,999,999
|January 7, 2003||
100,000,000,000 - 999,999,999,999
|January 7, 2003||
1,000,000,000,000 - 9,999,999,999,999
|January 25, 2003||
10,000,000,000,000 - 99,999,999,999,999
|March 31, 2003||Wade VanLandingham||Ben Despres||4,455,551|
100,000,000,000,000 - 999,999,999,999,999
|February 15, 2005||Matt Stenson||Ben Despres||49,436,290|
1,000,000,000,000,000 - 9,999,999,999,999,999
|February 2005||Matt Stenson||Ben Despres||52,964,177|
10,000,000,000,000,000 - 99,999,999,999,999,999
|July 2010||Matt Stenson||Ben Despres||529,181,042|
100,000,000,000,000,000 - 999,999,999,999,999,999
|December 2011||Matt Stenson||Ben Despres||606,337,405|
NOTE: I've been informed that some people feel that Ian Peters should be credited with the seeds up to and including all the 10 digit sets, based on the last section of Ian's Page. I don't disagree that it certainly appears that Ian did the work before Ben, but I have NOTHING from Ian to provide enough dates or data to feel like he is being "cheated" for his work. If Ian decides to write me and give me some dates, I'll reconsider the table above. I just wanted to make note of it here, for clarity that I have seen Ian's page, but don't feel it's enough info to change my own data.
NOTE: I have stopped searching for the 15 digit Seeds, after over 7 months. I have sent my file to Ben and Matt, and they are working on them now...
Number Range seems pretty clear to me. If you don't understand, write me. :-)
Date Complete - This is my best guess of when the last Lychrel in the range was found. I have the date stamps from Ben's files, but they may not be correct. (Ben, if you know different dates, let me know.)
Discoverer - Who's computer was running the app.
App Coder - As with my 196 search, I believe that the people who took the time to write the software should get as much credit as whoever was running the app. Maybe even more!!
Seeds Found - Is how many Seed numbers were discovered within the number range under test. It is NOT the total number of Seeds from 0 to the end of the range. It is also NOT the total numbers of Lychrel numbers in the range.
If you want any of the data files, most of them are available on the Files, Files, Files page....
Ben has managed to create a compression algorithm that is amazing!! For an example, the uncompressed file containing ONLY the 14 digit numbers is 217,557KB but using his compression program it comes out at 5,576KB and then using WinZip on THAT file, gives a file size of 3,694KB!!! To compare, if I just use WinZip on the original file, it only compresses to: 24,314KB!!! Amazing!
Something I want to point out. All of the numbers on the list, and all of the numbers in the chart above, are the SEED numbers. the Kin numbers like 295, 394.... etc do not show up on the list, nor do any other numbers that follow the same thread as any of the numbers that are on the list.
All numbers that do not become palindromes are Lychrel numbers. The smallest number in a thread is known as it's Seed number. All other numbers that converge onto the thread of a Seed number, are known as Kin numbers. Koji Yamashita had already named the related numbers as "Kin Numbers" in a paper of his from 1997, by the time these pages were written. For any further clarification, you can always refer to the Definitions page. For word usage, the Definitions page will always be more up to date than any other page on the site.SOME INTERESTING OBSERVATIONS ABOUT THE ABOVE TABLE BY BEN AND WADE>>>>>>>>>
6/29/02 From: Ben To: Wade
>> I don't know if you looked at the number of Lychrels in
>> a given number range, but it looks like this:
Hmm, no, I hadn't actually broken it down like that and considered it. Comparing against the hypothesis of Lychrels having a "random" distribution, we would expect to see nine times as many for N+1 digits as for N digits (since nine times as many numbers exist in that larger range). If anything, I would expect to see *less* than nine times as many, since at least *some* should converge with earlier series. Yet, we seem to have somewhere around 17 times as many, slightly less than 9*2 times more per order of magnitude. Obviously that trend cannot last, since at some point the number of Lychrels would overtake the number of integers (a fairly rigid upper limit ).
>>I don't know how many iterations you have taken these to,
>>but it seems to me that the larger the starting number is,
>>the longer it should take to form a palindrome.
First, I need to explain that I've used the AI technique of "iterative deepening", where I consider "number of digits" as the measure of depth. So, because of that, I can more quickly check a set of known possible Lychrels to a greater depth (For example, the ones I sent you I ran to 500 and took two weeks, but checked to 5000 overnight). I first check them to 11 digits (which weeds out numbers that very quickly reach a palindrome), then to 40 digits (which eliminates all but a very small number of stragglers), then to 500 digits. Between 40 and 500, fewer than 1% dropped out as Lychrels. Between 500 and 5000, not a single one did (and, as I mentioned for the 1E8 data, I ran those *much* further and not a single one dropped out from 500 to whatever-i-ran-them-to (100,000 digits?)).
So, this seems to imply rather strongly that the numbers I sent you, while not "proven" as Lychrels, will not fail as Lychrels for any digit length limit that we can reasonably test. I fully expect that *some* will eventually prove non-Lychrels, but a handful at most (assuming, of course, that Lychrels exist at all... It would certainly make us stop and think for a minute if 196 suddenly reached a palindrome, eh?
6/28/02 From: Wade To: Ben
I was thinking about this, and wanted your thoughts.....
I don't know if you looked at the number of Lychrels in a given number range, but it looks like this:
0 - 100 = 0
100 - 1,000 =2
1,000 - 10,000 =3
10,000 - 100,000 =69
100,000 - 1,000,000 =99
10,000,000 - 100,000,000 =1,728
100,000,000 - 1,000,000,000 = 29,813
A quick thought that I have had about the fact that there are so many more numbers for a higher range than a lower one was to wonder if they too will form out, but it will take a higher average number of iterations. (Is this the same as Jason's work?!?)
For example, if between 0 and 10,000, it takes an average of say, 5 iterations, to form a palindrome, and between 10,000 and 1,000,000 it takes an average of 60 iterations, (I'm just making up the numbers here) wouldn't it be logical that 100,000,000 to 1,000,000,000 would take say, 1,000 iterations or even 10,000,000?
For that matter, what *is* the average number of iterations that it takes to form a palindrome for the range 0-10,000 or 100,000,000 to 1,000,000,000? (There's ANOTHER program you can try to find time to write and I'll find a machine to run!!) :-)
I don't know how many iterations you have taken these to, but it seems to me that the larger the starting number is, the longer it should take to form a palindrome.
Either that, or as seems to be indicated on the chart and list that you provided, they will become more and more common, until eventually every number will be a Lychrel Number....
I don't know if you ever read that I changed the MSB of the 1 million data set, and ran it for 10 million iterations (ending archive entry 1/30/02), without a palindrome forming. I either hit a Lychrel Number by accident, or because they were more frequent, or because it required a far higher number of iterations to solve.