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Lychrel Numbers and Palindromes

11/23/05 There really is no need to explain my long dormancy on this site... I'm lazy, I've been busy, Florida springs and summers are meant for being outdoors, but really, I think it mostly boils down to the fact that nothing really new has happened. There have of course been events going on, but as far as radical new ideas or programs, it's all just been maintaining an even keel...

Some of the notable events since April...

Back on 11/6/05, Eric Goldstein passed 200 million digits for the Lychrel number 879!! Great job Eric!!! His status has been updated on the Lychrel Records page.

Felipe Barone has been forced to close the 196 forum. He has been recently plagued by hack attempts and member requests from spammers. Anyway, it never really took off like we expected it to, and he just doesn't have the time for the maintenance on it. Thanks for all your efforts Felipe!

Jason Doucette has finished his longest delayed quest search of all 18 digit numbers. As soon as I get permission from Jason, you will be able to see the results on his own World Records web page, or here on the Other People's Notes page. The reason for the delay is that Vaughn Suite also has calculated the same data and since Jason has not published his, and I have not shared it with Vaughn, they are in essence verifying the accuracy of Vaughn's app through a blind test.

The paper by Chris Pazsint, that I mentioned in the last update has been posted on the Math Solutions page. Chris really did a great job with this, and I thank him for his effort!

On 11/19/05, Pierre-André LAURENT from "France, working in Ireland" wrote to let me know that my FAQ is wrong where I answer the question: Does this have any use? with the response: Absolutely none. If you have a use, let us know. Otherwise, it doesn't really have any application. He makes his argument that it has validity with the 2 emails that I have condensed and put HERE. Thanks Pierre! It's nice to think that someday, someone will look at our work and put it to a practical application.

Ironically, the NEXT DAY (!!) I got an email from Jim Branson of the United States also discussing the reason for continuing our research into palindromes, 196 and Lychrel numbers in general. There was some information in the email that I didn't fully understand, and I have written him to ask for clarification, but he closed his note with a paragraph that I really enjoyed for some reason. maybe you will enjoy it as well:

One knows that dividing any number by seven, imparts a repeating nature to the result. I think we need to think about a lot of data as patterns and not only as quantities. If this seems to smack of "intelligent design" then one should consider the basic nature of "observation" that separates quantum mechanics from Einstein's world. It seems to me that we ourselves curve the very nature of the material world by what we think centuries from now. Is that God? I don't really think so. Invoking the nature of God seems to end intelligent thought and folks walk away from serious analysis.

And finally, Me??? What I have accomplished in the last 7 months?? Not much. But my computer is still plugging away, and all of the Milestones and File Verification entries have been updated. :-)

And just for goofy fun... Even though I haven't added any new visiting countries in the last few months, the Real Tracker reports that I've had 40,890 unique visitors to the site since I started the monitoring on August 4, 2002. Kinda cool.

4/28/05 Chris Pazsint dropped me the following bit of information concerning the new class of palindromes: I did come up with a name for numbers like 111, 131, 171, 191 etc I called them "Prime Palindromes". They are not hard to find it has to do with the odd number of digits and the center digit being odd. I have not quite worked out the finer details on the subject. I think XYX where Y is odd will turn out to not be the result of an add reverse. Simple reason behind it is odd+odd = even and even + even = even. The center number in an odd digit length number will always be added to itself.

Makes perfectly good sense to me.

Dan (no last name) wrote to vote for the name "Pazsint Palindromes" since they were discovered as a result of Chris' work. I'm O.K. with that.

So far, those are the only two contributions I've had. Send me more! Pazsint Palindromes is an interesting sounding phrase, but it's a mouthful to say, isn't it?!?! :-)

I know. I know... I'm probably making a mountain out of a molehill. But I really like the idea of naming this group of numbers, so one way or another, I'm going to. I get final say on this site, right? :-) :-) :-)

Syria has been added to the Where Were You list.

4/25/05 I've made a new discovery in MY world of knowledge!! Now, one of you can tell me that you've already thought of it, done it, or proven it or disprove me in this particular case...

Chris Pazsint of the United States wrote me a short while ago, and asked if I had ever figured out which thread the number 17509097067 comes from as mentioned on my Math Solutions page. He said he was going to take on the challenge of figuring out how to "Un Reverse and Add" a number. I told him I had not worked on it, and as far as I knew, neither had any one else.

Earlier today, I got a really elegant paper describing the process to "Un Reverse and Add" a number from Chris. It seemed a bit confusing to me at first, but I have spent almost the entire evening sitting here on my couch and working problems over and over again on my laptop, using his process. And even I have figured it out, and have managed to find what the possible Kin Numbers are for the thread that leads to the given number. It really is remarkable to me in it's simplicity of operation. I've asked Chris for permission to post it, so when I hear back from him, I'll get it here somewhere.

I'm sure some of you might find it a no brainer and nothing to get excited about, but it opened up a new area for me to play around with and occupy an otherwise unproductive Monday evening...

ANYWAY... While trying to make sure I understood the process, I was trying solve numbers that I knew what the answers SHOULD be so I knew if I had it right or not. For example, I solved the Kins leading up to 7,436 which is the third iteration of 196 and I solved the Kins leading to 52,515 which is the 5th iteration of 196. Stuff like that...

Then at random, I tried the palindrome 123767321. After trying and trying and worrying that I had a mistake somewhere, it occurred to me that the palindrome 123767321 CANNOT be created through the Reverse and Add process!!

As far as I can remember, this is the first time that I have ever considered whether all palindromes can be made through the RaA process. We have spent so many countless hours thinking about trying to get non palindromes to become palindromes, that I don't think it ever occurred to me to wonder if a palindrome HAD to come from the RaA process. This was a "breakthrough" in some part of my mind.

Naturally, the next thought I had down this path was "What is the smallest palindrome that can't be made through the Reverse and Add process?"

So some quick mental work, a double check with Chris' algorithm and I can't solve 111 from anything. (This is where you can disprove me!!)

But that in turn brought up another thought. One that we could actually waste EVEN MORE computer cycles on... How many palindromes are there that don't come from other numbers in the RaA process?? And which ones are they?? There are 10 that solve, before the first one comes up, (11, 22, 33, 44, 55, 66, 77, 88, 99, 101) and I found one at random, so I would expect there to be quite a few. I mean 131 is just two steps away. But maybe not and I just got lucky with the 123767321 number...

I am really excited that we have identified an entirely new sub-class of numbers!!! Just as "Seeds" and "Kins" are sub-classes under the broader class of "Lychrel Numbers", now we need a name for the "anti-lychrel" palindromes which can't be formed in our traditional methods!! I'm sure even Patrick De Geest's World of Numbers page would be interested in whatever we come up with...

I had the honor of creating the first new word on this site, so I would like to hear everyone's opinion of naming this new sub-class. I'm also open to ideas from everyone on how to "vote" for a name if I get more than one entry. :-)

I know there are a handful of schools and young students watching for updates on this site. Please, all of you send me your ideas, and maybe one of you young children can add a word to the world's languages!!

And for the record, thank you ahead of time, but no. I will not consider "VanLandingham Palindromes". The world does not want to spell my name for the next thousand years. :-)

4/22/05 I've been absolutely horrible on checking email and such, so to all of you who have written, and I didn't get back to you, I apologize. Spring is always a time I want to be doing things outside, and this year has proven no different. But by now, I think everyone probably understands that I do this in spurts of interest. I'll have long periods of nothing, then a few weeks of news, then a long period of nothing. The difference this time, is I feel I haven't even checked email enough to get timely responses to people. Sorry...

Then, Eric pointed out to me that the site was totally down for a week, and I didn't even notice it!! :-( Thanks Eric.

LOTS of odds and ends to clear up...

Jason Kruppa has re-emerged from the shadowy underworld of this project and is taking on some of the Lychrel numbers on some spare machines. When he gets me the numbers, I'll get them posted on the Lychrel Records page. (Assuming it doesn't take me another 3 months to do an update.

Ms. Sarah Feldt... I did actually try to reply to you within a couple days or so of your email, but it bounced, and I could never get back to you. Thanks for all you wrote. I appreciate it!!

Matt, vaughn and Eric have sent me a lot of historical data for their datasets, and I think I have them all in the right places over off of the Lychrel Records page!!

Jason Doucette was making some comments to me about the fact that both he and I have been unable to find any patterns in the 196 datasets Some of his comments are interesting:

My thoughts on this phenomenon are that it is very strange. If there is no pattern, it is essentially similar to an irrational number, where the digits are basically random. In other words, this pattern could be used to create truly random numbers. Consider that random numbers are difficult to create with a deterministic machine like a computer - basically impossible to make. Also, if it is truly random, then, it means there is a chance that the digits may line up properly to produce a palindrome, eventually. Yes, the chances are diminishing, but if it is random, there still exists that chance. Thus, it seems to be the case that it will eventually solve out, sometime before an infinite number of iterations. Note that 500,000,000 iterations may not be anywhere close to the required number of iterations.

Another thought which bugs me is that it seems as though the digits would have 'worked themselves out' by now. Judging by the results of the Most Delayed quest, when they do resolve, they resolve quickly. Perhaps this could be proven in 'reverse', by showing that the numbers have to resolve quickly, if they resolve at all. In other words, perhaps it could be shown that any such pattern that must occur for them to resolve, is resolved quickly. After all, there are only so many possibilities, and the Most Delayed quest is enumerating ALL of them. I am into the 18 digit numbers, now, having never missed a SINGLE number. That's quite a lot of numbers, and a lot of data that indicates this is the case. Just some thoughts...

Zimbabwe has brought the total number of visiting countries to 107.

And the most amazing news for this update is that Matt and Ben have sped up the Lychrel Seed program to phenomenal speeds!!! When I was running the original search app on a Pentium IV 1.7GHz machine, it took me over 2 months to run the 14 digit set, and since the 15 digit list of Lychrel Seeds was expected to be an order of magnitude larger, it was expected to take thousands of years for me to complete. As a result... I quit...

Ben's new app running on Matt's Athlon 64 went from 1 to completing the 15 digits in 47 hours!! STUNNING!!!

Matt finished the 15 dataset on February 15, 2005. Since he was able to verify the sets that Ben and I had previously come up with, it was discovered that I had made an error reporting the number of Lychrel Seeds in my original table. That error has been corrected, and the numbers that are there now have been verified by Vaughn.

Then they went on to completing the 16 digit set as well in only 6 days!!! I don't have some of the data yet, but what I do have is on the Lychrel Seeds page. Now there are 122,094,404 Lychrel Seeds identified. GREAT JOB guys!!

I still have more updates that I need to make, but it's taken me a couple hours just to get email caught up and these updates made to the various pages... I've still got Milestones, Dataset Info and File Verification info to get up as well as testing a new app version from Vaughn, and an entirely new app from Barry Roberts in British Columbia, Canada... I'm booked to spend most of tomorrow diving, so it probably won't happen tomorrow, but I'll try like hell to have the rest of these updates done by Monday 4/25...

THANKS for everyone's patience!!

2/9/05 Eric passed 170 million digits for 879, back on 2/5/05. Of course, I have updated the Lychrel Records page to show his progress.

There is a ton of email flying back and forth through all the key players, concerning compression and the Lychrel Seed searches. I would spend the rest of the day trying to condense the information that has been passed in the last couple weeks, and even then, I would misrepresent most of it, so I'm not even going to try. I would like to point out, again, that there are some extremely talented people within this 196 community, and they all have my deepest admiration for the work that they have done!!!


Of particular note, I would like to thank Eric Goldstein for the compression tool that he wrote. Backing up these massively huge files, is getting harder and harder, due to the fact that the files are getting so big. I am accumulating CDR's quite rapidly. Eric managed to get an incredible compression ratio. For example, the latest file at 234 million digits is 235,045 KB in size. If I use WinZip v8 on the file, I get a new size of 110,817 KB. Very nice. However, if I use Eric's app instead of WinZip, the file shrinks to only 70,113 KB!! Remarkable... In complete fairness, I'm not sure if the method was Eric's alone, or if it was based on input from Matt, Ben and Vaughn. I know all of them have been talking, I just don't remember how it actually came about. I applaud them all, and apologize if I have the credit wrong!!

Matt also wrote a small app that strips header information from all my files, and will save me hours and hours of time, when I do my site updates. Thanks Matt!!

Other than that... I've been watching the email quietly, and keeping in the background while these master artists paint there worlds... I can't wait to see what they come up with next!!

1/27/05 I have modified the Lychrel Records page quite heavily. Breaking it up into smaller more easily managed sections. I think it's nicer...

One modification I have made is that I have decided to not list the "Date Reported" for each of the verification sub-pages. It proved to be cumbersome for me, to get straight, and I didn't see where it added that much value. Instead, I changed the column to be the credited person with that milestone. I'm sure no one will mind the change, but if for some reason, you feel I should add that column back into place, let me know, and I'll consider it.

The "Date Result Reported" is still listed on the overall "Leaderboard" as the most recent data that I know of.

1/23/05 You can see I've archived everything prior to 1/1/05, so if there is something old that you're looking for, head over to the Blackboard Archive.

Vaughn has started running some of the other lychrel numbers on various machines he has access to. He says in his email to me: I would like to announce that I have started processing 9999, 99999, 999999, 9999999, 99999999, and have taken all to 2 million digits or above as following. Please note that all seem to follow different paths since the first 2 million digit number of each is different. I intend to take them all to 20 million digits in the first instance, and thereafter one by one to 100 million digits. I intend to use a 2.66GHz machine with a 533MHz memory bus, so it will be severely memory bound.

I have added his information to the Lychrel Records page.

Vaughn also provided the verification data for each file, similar to what I have done for 196 on the File Verification page. I thought it was a great idea, so I have asked Matt and Eric to send me the data for their files as well. I asked them to only send me the data for every 25 million digits.

So for now, I have broken the Lychrel Records page into two sections. The Identified Lychrel Seeds page shows how many digit sets have been searched and the number of known lychrel numbers found so far. The Lychrel Records page has changed to only show the non 196 seed numbers that are being tested using a similar "brute force" method to what I'm doing.

Wandering the web earlier today, I stumbled across a 196 implementation that I found quite funny. Apparently, there is a programming language with the really funny name of BRAINFUCK that can be read about in the following Wikipedia entry. Well, Mats Linander took the time to write a 196 program in this oddly titled language! With all the commenting removed, it looks like this:

>>>>>>>[<< [->>[>>>>>]<+<<<<[<<<<<]>>>]>>[>>>>>]<<[-<<<[<<<<<]>>>+>>[>>>>>]<<]

I THINK I only removed the comments. I may have deleted some data as well, so if your interested in reading Mats' original writeup, you can see it at http://esoteric.sange.fi/brainfuck/bf-source/prog/196-commented.b. And if the link stops working for whatever reason, there is a local copy here. I just found it interesting, so I made note of here. Maybe Ben will add a copy of it to his site! :-O

1/18/05 My apologies to Vaughn!! Yesterday, I said: He's very busy lately with the first birthday for his son coming this month, but... Well, he wrote and let me know that his DAUGHTER will be happy to celebrate her birthday on 1/25/05. I goofed... I hope she's not reading this!! :-)

Matt sent updates for his searches with 1997 and 10877. He's passed the 106 and 42 million digit mark respectively. It was nice, because I haven't had an update for 10877 since 9/16/03!! Of course, it's now all updated on the Lychrel Records page.

1/17/05 Man, It's been a while since I updated this site, and many of you have written to let me know!! Sorry.

We all know how it is... You get doing something else, and the newer task becomes more interesting than the one that you have been doing for a while. Maintaining this site, and "studying" this problem is something I've done for quite a while now, and it just gets "boring" sometimes, when compared to other things I could be doing. As a result, this site suffers.

Not that it matters to anyone, but just to give you an idea of what has been absorbing my time: I took a trip to England for the entire Christmas - New Years holiday period. That was a great time. I learned to SCUBA dive back in October, with my girlfriend and we've been trying to do that as much as possible. (Even going so far as to dive the English Channel on 12/27/04!! It was MUCH, MUCH colder than the Florida water that we've been used to!!! :-) ) I modified my milling machine to operate as a CNC machine. And that project alone has kept me in the workshop almost every evening for the last 2 months. I made my first drawings with the machine only recently on 1/14/05, by using a pencil instead of an end mill in the cutter. So it's working, and now it's a matter of cleaning up all my parts and circuits. It turned out to be a MUCH bigger task than I had originally planned!! Anyway... That's what I've been doing. The downside, is that now, it's taken me most of an afternoon to update this site!!


The simple stuff: Cuba, Ecuador, Kuwait, Sri Lanka, Venezuela and Yemen have all been added to the visiting countries list, bringing the total to 106 countries.

Eric passed 160 million digits for 879, way back on 11/25. I would speculate that he has passed 170 by now, but I haven't confirmed it with him.

On December 15, Jason Doucette completed the longest delayed palindrome search for all 17 digit numbers. The output information from his program is over on the Other People's Notes page. Congratulations to Jason for pushing that quest to the limits he has!! His page is Here.

Eric sent me a new version of his app back on 11/6/04, but due to my goofing around, I just got to testing it today. It's interesting... On Eric's machine, the last three updates he's done (v253, v254 and v255) have all shown speed increases for him, but when I run them on my machine, they are all slower than v252.1 that I have been using since Oct of 03. They are faster compared to each other (ie v255 is faster than v254 which is faster than v253), but the v255 is still not as fast as v252.1 on my machine... There must be an explanation for it, but one of you programmer type people will have to figure it out. It's beyond me.

Vaughn Suite has been busy as well... On 12/30/04, he sent me his latest jump into the pool. His latest app did indeed run faster for me than the last one he sent. He has cut 1 minute and 18 seconds from his previous best deep test time!! The new results are posted on the Software Comparisons page. Great job to Vaughn!!! He's very busy lately with the first birthday for his son coming this month, but I expect he'll keep squeezing every last cycle out of my CPU.

And finally, I wish everyone a safe and happy 2005!!!