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World Records - Palindromes

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Record Attempt Time Commenced Personal Achievement
1. 196 Palindrome Quest August 4, 1999 Former WORLD RECORD achieved September 6, 1999.
Current WORLD RECORD achieved by Wade VanLandingham.
2. Most Delayed Palindrome August 13, 1999 Former WORLD RECORD achieved September 3, 1999.
Current WORLD RECORD achieved November 30, 2005.

What is a Palindrome?

A palindrome is something that reads the same forward as it does backward. It originated in the early 17th century from the Greek word palindromos (palíndromos), literally meaning "running back again."

Numbers:
Words:
Phrases:
DNA Sequence:
"52125", "4334", "8", and "1758571".
"Radar", "I", "Eve", "Deed", and the world's longest in the English language: "Redivider".
"Madam, I'm Adam", and the timeless classic: "A man, a plan, a canal... Panama".
A segment of DNA in which the nucleotide sequence of one strand mirrors that of the complementary strand.

(The world's longest palindromic sentence, based on the timeless classic, "A man, a plan, a canal... Panama", has been created by Peter Norvig. For more word, sentence, and phrase palindromes, please visit Jim Kalb's Palindrome Connection.)

196 PALINDROME QUEST

Numeric Palindromes

In the April 1984 Scientific American "Computer Recreations" column, an article appeared about mathematical patterns (F. Gruenberger, Computer Recreations, "How to Handle Numbers with Thousands of Digits, and Why One Might Want To.", Scientific American, 250 [No. 4, April, 1984], 19-26.). Here's the algorithm:

  1. Pick a number.
  2. Reverse its digits and add this value to the original number.
  3. If this is not a palindrome, go back to step 2 and repeat.
Do all numbers eventually become palindromes by this process? It was suggested that this is the case.

Examples

Most numbers become palindromes fairly quickly, in only a couple of steps:

13
  1. 13 + 31 = 44



64
  1. 64 + 46 = 110
  2. 110 + 011 = 121


87
  1. 87 + 78 = 165
  2. 165 + 561 = 726
  3. 726 + 627 = 1353
  4. 1353 + 3531 = 4884

In fact, about 80% of all numbers under 10,000 solve in 4 or less steps. About 90% solve in 7 steps or less. A rare case, number 89, takes 24 iterations to become a palindrome. It takes the most steps of any number under 10,000 that has been resolved into a palindrome:

89
  1. 89 + 98 = 187
  2. 187 + 781 = 968
  3. 968 + 869 = 1837
  4. 1837 + 7381 = 9218
  5. 9218 + 8129 = 17347
  6. 17347 + 74371 = 91718
  7. 91718 + 81719 = 173437
  8. 173437 + 734371 = 907808
  9. 907808 + 808709 = 1716517
  10. 1716517 + 7156171 = 8872688
  11. 8872688 + 8862788 = 17735476
  12. 17735476 + 67453771 = 85189247
  13. 85189247 + 74298158 = 159487405
  14. 159487405 + 504784951 = 664272356
  15. 664272356 + 653272466 = 1317544822
  16. 1317544822 + 2284457131 = 3602001953
  17. 3602001953 + 3591002063 = 7193004016
  18. 7193004016 + 6104003917 = 13297007933
  19. 13297007933 + 33970079231 = 47267087164
  20. 47267087164 + 46178076274 = 93445163438
  21. 93445163438 + 83436154439 = 176881317877
  22. 176881317877 + 778713188671 = 955594506548
  23. 955594506548 + 845605495559 = 1801200002107
  24. 1801200002107 + 7012000021081 = 8813200023188

The 196 Palindrome Quest (a.k.a. 196 Algorithm, 196 Problem)

Does every number eventually become a palindrome? Nobody knows for sure, since it has never been proven. There are some numbers that do not appear to ever form a palindrome. The first one is 196. Such numbers are called Lychrels. The search to resolve this number has been referred to as the 196 Algorithm or the 196 Problem, but normally called the 196 Palindrome Quest.

There are a few references to the 196 Palindrome Quest before the 1984 issue of Scientific American. Two references were found from a Math Central page:

The 1973 paper states that Harborth said that Trigg checked all integers less than 10,000 in 1967 and found that 249 seemed to never form a palindrome. 196 would be the first of those 249 numbers.

John Walker's Three Years Of Computing page states that 196 had originally been iterated over 10,000 times without yielding a palindrome. Later, this number had been carried through 50,000 reversals and additions by Paul C. Leyland yielding a number of more than 26,000 digits without producing a palindrome. Paul explains his program in an email to Wade VanLandingham on August 19, 2002:


"The work you refer to was done about 20 years ago on a 4 MHz Z80-based machine running CP/M. The core reverse&add and the palindromicity detector were written in assembler and the I/O etc was written in Algol-60. The machine only had 32K of memory (actually rather a lot for those days) and I ran the program until it ran out of memory --- which explains the limit chosen for the number of iterations." - Paul C. Leyland


Again, P. Anderton continued the process up to 70,928 digits (170,000+ iterations) without encountering a palindrome. (This Google Groups posting claims a 70,000 digit result was accomplished in 1987, so perhaps this was P. Anderton's effort.)


The First Iterations of 196

Here is 196 iterated 200 times for those of you who are curious.


Three Years of Computing (196 taken to 1,000,000 digits)

On August 12, 1987, John Walker commenced a program he created on a Sun 3/260 workstation (using a Motorola 68020 microprocessor) to take the search even further. Five minutes before midnight, on May 24, 1990, almost three years later, his program ended after 2,415,836 iterations, yielding a number 1,000,000 digits long - with no palindrome in sight. The next day, John Walker made this 1,000,000 digit number and his program available to people on a page on the Internet, Three Years Of Computing, for any who would like to continue his quest without redoing his three years of work.


About Two Months of Computing (from 1,000,000 to 2,000,000 digits)

With a interest in the palindrome quest since the 1984 issue of Scientific American, Tim Irvin found himself with access to a supercomputer, a Concurrent Computer Corporation Maxion model 9502, in 1995. Searching the Internet for work already completed on the palindrome quest, he found John Walker's web page. He 'borrowed' his program, and with the help of fellow programmer Larry Simkins, they put it to work again to continue the quest from the 1,000,000 digit number. On August 22, 1995, after about two months of calculations, the program stopped after having calculated 196 to 2,000,000 digits. With a difference of 8 years in computer generations, Tim's supercomputer achieved three times the amount of calculations (as required for John Walker's three years of work) in only two months. This number is available off of Tim's web page, About Two Months of Computing, for any person wishing to continue the quest themselves.


Continuing the Quest (to 13,000,000 digits and beyond)

I have never even seen a supercomputer, so why am I continuing the quest? Because I created an assembly language program that can perform the necessary computations on an average desktop computer faster than the unoptimized programs previously used on supercomputers can. (Please note that the original programs created to compute the 196 Palindrome Quest were made to be memory efficient, at the sacrifice of speed. Despite these concerns, I found that these programs missed certain optimizations that could have saved the same amount of memory with no speed sacrifices, as well as certain optimizations that had no bearing on memory use whatsoever).

My program started calculating the 196 Palindrome Quest on Monday, August 9, 1999 on a Pentium II 266 MHz PC.

The program reached the 1,000,000 digit mark in 1 day and 18 hours

I am going to retest this time at some point, since my code was bugged, causing it write a total of 54 megabytes of junk to the screen and a data file, which slowed the calculations.

The program reached the 2,000,000 digit mark in an additional 5 days and 10 hours.

Both these numbers check out with John Walker's and Tim Irvin's work. I am 100% confident that my program is bug-free, I am going to continue the work for as long as I have access to idle time of a computer.

The program reached the 3,000,000 digit mark in an additional 8 days and 7 hours.

The program reached the 4,000,000 digit mark in an additional 8 days and 14 hours. This was achieved on Monday, September 6, 1999, breaking the previous record of 3.9 million digits (according to the rec.puzzles archive), 4 days quicker than expected, thanks to the help of a Celeron 400 MHz machine, on which the program ran for some of its processing.

The program reached the 5,000,000 digit mark in an additional 13 days and 8 hours.

The program is now running solely on the Celeron 400 MHz processor.

The program reached the 6,000,000 digit mark in an additional 14 days and 4 hours.

The program reached the 7,000,000 digit mark in an additional 26 days and 12 hours.

The program reached the 8,000,000 digit mark in an additional 33 days and 23 hours.

The program reached the 9,000,000 digit mark in an additional 59 days and 12 hours.
(The increased length of time for this million digits is probably caused by the fact that the program was stopped and continued very many times during this period.)

The program reached the 10,000,000 digit mark in an additional 34 days and 1 hour.

The program reached the 11,000,000 digit mark in an additional 26 days and 3 hours.

The program reached the 12,000,000 digit mark in an additional 28 days and 5 hours.

The program reached the 13,000,000 digit mark in an additional 28 days and 23 hours.
(The program had a little help of a Pentium III 450 MHz machine for this million digits, although the program can only be run on one machine at any given time.)

My 13,000,000 digit mark is published in the November / December 2001 Issue of Yes Mag: Canada's Science Magazine for Kids:

Page 16, Nov/Dec 2001 Issue, Yes Mag: Canada's Science Magazine for Kids
Click to enlarge [75 dpi resolution image = 154 Kb]

Click to enlarge [150 dpi resolution image = 452 Kb]

The program was averaging over 12,000,000 single digit additions per second with the Celeron 400 MHz machine (i.e. it can add two 12,000,000 digit numbers in one second). That's over 1 trillion (1,000,000,000,000) single digit additions per day.


Other Work - Ian J. Peter

After my correspondence with Ian J. Peter, who is currently calculating all integers up to 2,000,000,000 out to 200,000 digits each, he started to improve his palindrome program. His hand optimized assembly program, running on an AMD Athlon 500MHz system runs from 4 to 8 times as fast as my program on an Intel Celeron 400MHz.

Why such a huge speed increase? Several factors. His program has been largely optimized algorithmically, as well as in hand optimization at the assembly level - but I am not going to give out any secrets here; it is not for me to release, so please ask him instead. But I can say a few technical notes:

My program uses flat-mode (unreal-mode) 32-bit addressing in 386-code, using Turbo Pascal 7.0, due to my lack of a compiler that allows true protected mode memory addressing. This is amusing, given that the Intel 80286 (i.e. 286) processor introduced this feature (My old Tandy 1000 SX had one of those). Ian is using protected-mode 32-bit addressing, running under Linux, and is taking advantage of the new instruction set on the newer machines.

As far as hardware is concerned, this is the first time AMD has impressed me. I have an AMD K6-200MHz, which I am not impressed with. My 3Dfx VooDoo2 card sits and waits for the processor to hand it information. But, the Athlon is another story. It has 256 KB L1 cache (more L1 cache than Celerons have in L2 cache), which is 4 times the amount of regular Pentium III systems. It has 512 KB L2 cache, but can have as much as 8,192 KB. This does not mean you should go out and buy one. You should buy the processor that will benefit your type of work most. The nature of our palindrome programs - given that the code is very small, and that memory access is always in sequential order - benefit from the Athlon architecture even more so than normal applications or games.

His program reached 1,000,000 digits in just over 5 hours, and reached 10,000,000 digits in just over 30 days, running on his dedicated 500 MHz Athlon.


Other Work - István Bozsik

István Bozsik found the same previous work from other people, mentioned above, that I did. It sparked his interest, and he quickly found that he could also make a program to run the 196 Palindrome Quest much faster than they did. He wrote his program on May 7, 2000.

On his web page he shows that he took the quest to 6 million digits. Unfortunately, I had already passed 6 million digits with my own work. István had not been able to locate my web page via search engines until after his achievement. His page has lots of links to other web pages regarding palindromic numbers.


Work Passed On - Wade VanLandingham (to 300,000,000 digits and beyond)
I no longer have the processing time to continue the quest. I have given my 13.5 million digit record to Wade VanLandingham. He has taken this number, and continued it to over 300 million digits (almost 725 million iterations). He has compiled a lot of information from all kinds of sources regarding the 196 Palindrome Quest on his web page, 196 and Other Lychrel Numbers. It is worth a visit.


196 Discussion - Message Board

[Note: This message board is offline. If anyone is willing to host this message board, so it can continue to exist, please contact me.] Felipe Barone has created a message board for discussion the 196 Palindrome Quest. You may like to take a look around. I have some interesting thoughts regarding computation of the 196 Palindrome Quest on a network in the following thread: Processing Across a Network. [Internet Archive of Processing Across a Network thread.]

FRIDAY, JANUARY 27, 2006 UPDATE:
It has been done! Pierre-André Laurent has created a networked 196 Palindrome Quest application: 196 and Other Lychrel Numbers - Distributed Software Comparisons

MOST DELAYED PALINDROMIC NUMBER

The Previous Record

With the exception of my work, the only information on the Most Delayed Palindromic Number was found at Ian J. Peter's website: Search for the Biggest Numeric Palindrome.

Before his work, the most delayed palindromic number that was known was 10,911. It takes 55 iterations to become a palindrome that is 28 digits long.

After his extensive searching on all numbers from 1 to 9,999,999, he has found the following results:

Previous Records - found by Ian J. Peter
Number Iterations Resulting Palindrome
147,996
150,296
1,000,689
1,005,744
1,017,501
7,008,899
9,008,299
58
64
78
79
80
82
96
8834453324841674761484233544388
682049569465550121055564965940286
796589884324966945646549669423488985697
796589884324966945646549669423488985697
14674443960143265333356234106934447641
68586378655656964999946965655687368586
555458774083726674580862268085476627380477854555

The Procedure

Ian Peter tested all of his numbers to 200,000 digits (~500,000 iterations). He proved that there is no number from 1 to 9,999,999 that forms a palindrome in over 96 iterations, but in under ~500,000 iterations.

Using this information, I set my program (the same program currently trying to break the 196 Palindrome Quest record) to retrieve the number of iterations required to form a palindrome for all numbers from 0 to 9,999,999. This was done by setting a limit of 96 iterations. If the number did not resolve into a palindrome after this many iterations, it was marked as 'infinite'. Because of Ian Peter's work, I know that I would have to take these numbers to at least 500,000 iterations before resolving them. Knowing that the likelihood of this happening is very slim, I decided not to take them any further.

After analyzing the information, I determined that the chances of the next most delayed palindromic number being over 255 iterations, without first finding one that resolves in fewer iterations, was extremely unlikely.

As a result of this, I determined the best way to break the record of the Most Delayed Palindromic Number was to continue from 9,999,999 on, looking at a maximum of 255 iterations. By limiting the number of iterations to such a small number, an unbelievable amount of time is saved (If I wished to double the iteration limit to 510, the process instantly becomes four times slower).

Using this method, within the first two days of running the program, my program had solved two new records: 100,239,862, which solves in 97 iterations and 140,669,390, which solves in 98 iterations. However, I did not realize until after speaking with Ian J. Peter that his program had already solved these records, and he simply had not updated his web page to show the results.

Here are Ian J. Peter's latest results:

Previous Records - found by Ian J. Peter
Number Iterations Resulting Palindrome
100,239,862
140,669,390
1,090,001,921
1,009,049,407
1,050,027,948
1,304,199,693
1,005,499,526
97
98
99
101
104
105
109
1345428953367763125675365555635765213677633598245431
1345428953367763125675365555635765213677633598245431
6634544448788301675886446885761038878444454366
1543434266587555114779722279774115557856624343451
5831124885795990016569666669656100995975884211385
5831124885795990016569666669656100995975884211385
66330069478378985774345546664554347758987387496003366

Thoughts on Improving the Search

The current procedure is not the fastest way to find the most delayed palindromic number. Most numbers being checked are 'consequences' of numbers already checked. For instance, a few consequences of 140,669,390 are: 150,669,380, 160,669,370, 170,669,360, 180,669,350, 190,669,340. These were found by simply changing the second and the second-to-last digits. Note the pattern - each of these numbers, after their first iteration will all become the same number (the two digits being modified sum to 13 each time), and thus will eventually all yield the same palindrome. It does not make sense to recalculate these same iterations over and over again. Even more consequences can be found by changing the third and the third-to-last digits, and so on. These are all known as 'first order consequences' since they produce the same number after just one iteration.

Other consequences exist that produce the same number after two iterations (these are called 'second order consequences' since they produce the same number after two iterations), and after three iterations ('third order consequences'), and so on. Although, these are a little harder to find. If you'll look at Ian Peter's work in the table above, you will notice some of his results form the same palindrome as others. They are consequences of each other.

By messing with the digits, as we did above, we can find 1 * 6 * 4 * 4 = 96 numbers, including the original, that all have the same properties as 140,669,390. Almost every number, not just ones that become palindromes, have lots of first order consequences of itself - and in most cases, more than just 96.

However, until I write a program to fix these problems, the program will continue to run on its machine, for as long as I have access to it. It is currently calculating 100,000,000 numbers per day - so, hopefully we have seen just the beginning.


Improving the Search

FRIDAY, AUGUST 20, 1999 UPDATE:

BAD NEWS: I have lost access to the computer that was dedicated to these calculations.

GOOD NEWS: After a little bit of thinking, I have 'discovered' an algorithm in which will minimize repetitive calculations that are caused by first order consequences of the same number. This is an exponential improvement on my previous algorithm. Allowing my computer 2 weeks of work with my old algorithm, at its current rate, would 'solve', to 255 iterations, all numbers from 0 to 1,360,000,000. In others words, all 9-digit numbers and more. Theoretically, my new algorithm, given the same number of calculations (2 weeks of computations on the same machine), could 'solve', to 255 iterations, all numbers from 0 to 99,999,999,999,999. In others words, all 14-digit numbers. (My previous algorithm would take 2,800 years to do this) I will probably have to increase my iteration limit as the numbers grow large, thus taking more time than just 2 weeks, but the above is just to give an indication of the improvement of the algorithm itself. Enough talking - in a few days I am going to implement the algorithm, and hopefully it lives up to what I expect. I will run it on my home computer, which should 'see' about 50% of the processing that it was getting on the dedicated computer I used to have.


Implementing the Search

FRIDAY, SEPTEMBER 3, 1999 UPDATE:

I have reprogrammed my new exponential algorithm to find the smallest number that solves into a palindrome for each distinct iteration count. For example, it will find that 10,905,963 solves in 71 iterations, regardless that there is a smaller number that solves in more iterations (9,008,299 in 96 iterations). 10,905,963 is reported because it is the smallest number that resolves in 71 iterations, out of all numbers that resolve in 71 iterations. Thus the program not only reports numbers that break the record for more iterations, but also reports the smallest number that solves in x iterations, for all x.

The program adjusts the iteration limit so that it is three times that of the current record. This was programmed after analyzing the data already accumulated, and it was determined that the chances of finding a number that solves in more than three times the number of iterations of the current record, without first finding one that solves in less than three times the number of iterations of the current record, is extremely slim.

I have started the new program, calculating all most delayed palindromic numbers from scratch on Friday, September 3, 1999 on my AMD-K6 200 MHz computer.

FRIDAY, SEPTEMBER 3, 1999 UPDATE:
Within an hour and a half (1 hour, 29 minutes, 45 seconds, to be exact) of started the new program from scratch, it had solved a new world record:
The 11 digit number 10,287,799,930 resolves into a 65 digit palindrome in 134 iterations.
Amazingly, only 3 minutes later, a new world record was found:
The 11 digit number 10,700,572,940 resolves into a 66 digit palindrome in 149 iterations.

FRIDAY, AUGUST 4, 2000 UPDATE:

More statistics of the program follows:


Jason's New Exponential Algorithm
Performance (AMD-K6 200 MHz)
It solved all 6 digit numbers and below in 1 second.
It solved all 7 digit numbers in 28 seconds.
It solved all 8 digit numbers in 1 minute 38 seconds.
It solved all 9 digit numbers in 21 minutes 51 seconds.
It solved all 10 digit numbers in 59 minutes 16 seconds.
It solved all 11 digit numbers in 20 hours, 29 minutes.
It solved all 12 digit numbers in 1 day, 20 hours.
It solved all 13 digit numbers in 31 days, 9 hours.
It solved all 14 digit numbers in 63 days, 9 hours.
It solved 8.34% of all 15 digit numbers.


FRIDAY, AUGUST 4, 2000 UPDATE:
During this, it solved the following world records, each beating the last record my program set:

New World Record: The 13 digit number 1,600,402,195,090 resolves after 188 iterations.
New World Record: The 15 digit number 107,000,020,928,910 resolves after 192 iterations.
New World Record: The 15 digit number 100,120,849,299,260 resolves after 201 iterations.

At this point, the program stopped due to lack of a computer to run the program.



The Results

MONDAY, JUNE 17, 2002 UPDATE:

I have not been able to run my program since Friday, August 4, 2000. At the time, I only had 8.34% of all the 15 digit numbers solved. I could not continue because the program was written in DOS under Flat Mode, which refuses to run under Windows 98 or greater. These operating systems no longer allow the computer to be (easily) rebooted into a fully DOS compatible mode, which is not a Windows Shell.

On Sunday, May 19, 2002, I finally reprogrammed my algorithm in Windows to allow the search to continue. It is too bad that I have waited this long, because I would be far into the 17 digit numbers right now.

After some initial beta-testing, I started the new program again from scratch on Thursday, March 30, 2002. It has since reached, and surpassed the numbers checked of my old DOS program. Note that it is only running part time on my AMD 1700+ XP processor.

The records, including the World Record for Most Delayed Palindromic Number (highlighted in red), are listed in the below table. Click on any of the numbers to pop-up a calculation page that proves its result.

WEDNESDAY, APRIL 16, 2003 UPDATE:

I am now into the 17 digit numbers. This is going to be an interesting set of numbers to look at, since after the result of the 16 digit numbers, there is only one iteration count below the current (as of April 16, 2003) world record of 201 iterations (the 15 digit number 100,120,849,299,260) that has yet to solve out: 160. Any new information to be reported by the program will either be the 160 iteration result, or a new world record.

We can guess at what the 160 iteration result would be, by iterating the smallest number to resolve in 161 iterations (the 16 digit number 6,000,000,039,361,479) a single time. Thus we can tell that the 17 digit number 15,741,639,339,361,485 will solve out in 160 iterations. Its lowest first order consequence ('consequences' are explained above on this page) is 14,200,033,399,975,995. Therefore we will expect to see a result for 160 iterations which is, at largest, this 17 digit number.

TUESDAY, APRIL 29, 2003 UPDATE:

I have incorporated Benjamin Despres' reverse-and-add code into my algorithm for finding Most Delayed Palindromic Numbers, as his code is approximately three times faster than my reversal-addition code. The core exponential algorithm that determines which numbers to test remains unchanged. I have started the quest from scratch, again, to ensure that his code is operating perfectly with my code. I have already extensively tested his code with extreme cases to ensure its accuracy. The biggest problem that I had porting his code into my program is that his code was not designed to be re-run multiple times. He programmed the initialization portion of it for a one-shot deal. Initialization code is normally left unoptimized as it is only run once, and therefore does not matter for performance. It only affects the start up time of a program, and for a process that takes several months (his code was computing the 196 Palindrome Quest), it does not matter if the initialization code takes 1/1,000th of a second or 1/10,000th of a second. I re-coded the initialization section to re-initialize only the parts of memory that has to be for optimal performance.

WEDNESDAY, JUNE 2, 2003 UPDATE:

My old code had solved 0.297% of all 17 digit numbers. My new program (with Benjamin Despres' reversal-addition code) finally surpassed this today. All results match prior results. A few hours later, my program also finally found a new world record since the old record was found solving the 15 digit numbers. My old program was within hours of solving this new record, and was delayed for almost two months due to the restarting of the program with Benjamin Despres' reversal-addition code.

WEDNESDAY, JUNE 2, 2003 UPDATE:
Today, at 2:47 PM, my program (with Benjamin Despres' reversal-addition code) solved a new world record:
The 17 digit number 10,078,083,499,399,210 resolves into a 112 digit palindrome in 233 iterations.

This new world record beats the old world record that has stood for about 3 years (set by my program sometime between November 1999 to August 2000; the exact date was never recorded) which was the 15 digit number 100,120,849,299,260 which resolves into a 92 digit palindrome in 201 iterations.

Also, later that day, at 7:28 PM, a record for 160 iterations was found: 10,019,017,999,499,510. It is lower than the number that we expected could be the potential record (14,200,033,399,975,995), and it is on a different thread (series of numbers that are formed by the reversal-addition iteration), as the resultant palindrome is different from that of the record for 161 iterations. (Please note that on January 28, 2004, an even smaller number for 160 iterations was found, on yet another thread: 10,000,000,730,931,027.)

THURSDAY, JULY 10, 2003 UPDATE:
Today, at 12:49 PM, my program (with Benjamin Despres' reversal-addition code) solved a new world record:
The 17 digit number 10,442,000,392,399,960 resolves into a 111 digit palindrome in 236 iterations.

WEDNESDAY, DECEMBER 15, 2004 UPDATE:

At 10:44 AM, after exactly 500 days of processing from scratch, my program (with Benjamin Despres' reversal-addition code) finally resolved all 17 digit numbers. The majority of that time (444 days of those 500) was required for just the processing of the 17 digit numbers. Due to the optimizations in my algorithm, it does not iteratively check each number. My algorithm determines which numbers can be eliminated from the search and still maintain 100% accurate results. As a result, my program actually only checked 186,819,193,449 total numbers, instead of 99,999,999,999,999,999 numbers, to compute the results of all 17-digit numbers and smaller, which is a significant optimization. It would have taken over 700,000 years, on the same computer, to compute all these numbers without this optimization.

MONDAY, MARCH 21, 2005 UPDATE:

Vaughn Suite has corresponded with me regarding my quest over the past few weeks. He suggested performing a statistical analysis of the data, to determine the likelihood of missing a record at any given iteration limit. Using such analysis, we can set the limit to match our preferred risk. If this shows the current limitation is overkill, then we will speed up the program by reducing the limit.

Originally, I took a naive approach and simply set the limit to be 3 times that of the last record found. This was done only because, at a quick glance, it appeared to suffice. It appeared quite a bit less safe than Ian Peter's limitation of 200,000 digits (about 500,000 iterations, which varies slightly depending on the number checked). I should again note that Ian Peter's work has exhaustively shown that numbers that do not resolve quickly seemingly never resolve. This has been crucial in allowing us to use the below statistical analysis on my data.

Vaughn Suite's analysis started with the most delayed palindromic number for each digit length, as follows (updated Sunday, September 25, 2005 upon completion of the 18-digit set):

Largest Delay per Digit Set
Digit Set Largest Delay
1-digit numbers
2-digit numbers
3-digit numbers
4-digit numbers
5-digit numbers
6-digit numbers
7-digit numbers
8-digit numbers
9-digit numbers
10-digit numbers
11-digit numbers
12-digit numbers
13-digit numbers
14-digit numbers
15-digit numbers
16-digit numbers
17-digit numbers
18-digit numbers
2
24
23
21
55
64
96
96
98
109
149
149
188
186
201
197
236
232

You will note that this data is not immediately known from the records I show on this page. My Most Delayed Palindromic Number quest stores only the smallest number that resolves at each iteration depth. Therefore if a record is set in a smaller digit set, any other numbers that resolve in the same amount of iterations from larger digit sets will not be recorded, since they are larger numbers. These numbers are not recorded even if they are the most delayed palindromic number of that digit set, simply because this was not the purpose of the program.

For example, in the 7 digit set, the most delayed palindromic number is 9,008,299 which resolves in 96 iterations. There are two (consequences not counted) 8 digit numbers that solve in 96 iterations: 15,002,893 and 15,059,593, but they are not recorded. The most delayed palindromic number recorded for the 8 digit set is 10,309,988 which solves in only 95 iterations. The two that resolve in 96 iterations are not smaller than the 7 digit number, 9,008,299, that resolves in 96 iterations, and my program saves only the smallest one.

Vaughn Suite verified these maximum iteration depths for each digit set himself for all digit sets up to 13 digits, and got the maximum depths for the 15 digit and 17 digit sets from this page, and the for the 14 digit and 16 digit sets from my notes on Wade VanLandingham's site (under Other People's Notes). He noted that the values appeared to follow a linear trend, estimated the maximum iteration depth for the 18 digit set would likely be 250 and suggested testing to 350 iterations, instead of multiplying the last previous record by three. After some discussion, we performed linear regression analysis, which yielded the following equation:

Expected Maximum Iteration Depth = 14.416667 * Digit Length - 18.338235

The standard deviation was found to be 11.245233. We concluded that we could use normal distribution to describe the data. There is a 98.83% correlation between these iteration limits and the digit length of the set they represent. (If you are a statistician, and believe this is incorrect, please contact me.) Armed with this information, we did further spreadsheet analysis to determine the probability of a number being missed for the 18 digit set, giving any iteration limit. We could also input the probability (risk) level we were prepared to accept, and the sheet would inform us of the iteration level required for such a risk.

In retrospect, I was doing far more calculations than I needed to. Using a limitation of 708 iterations for the 18 digit set represents a missed record would have to be 41.51 standard deviations away from the mean. This is an astronomically small chance. I cannot even calculate the chances within Excel due to the precision limitations of the program. (If someone has Maple, Mathematica or another Computer Algebra System, please contact me, and perhaps we can compute the actual amount.) To give you an idea just how crazy this is, 68% of all numbers fall within 1 SD (standard deviation) of the mean, 95% fall within 2 SD, 99.7% fall within 3 SD, 99.993% fall within 4 SD, 99.99994% fall within 5 SD, 99.9999998% fall within 6 SD, 99.9999999997% fall within 7 SD, etc. In another perspective, if we wish to have a 1 in 1,000 chance of missing a record, we'd set the iteration limit 3.09 SD away from the mean. To have a 1 in 1,000,000 (one million) chance of failure, we'd set the iteration limit 4.75 SD away from the mean. For 1 in 1,000,000,000 (one billion), set it 6.00 SD away. For 1 in 1,000,000,000,000 (one thousand billion), set it 7.04 away. For 1 in 1,000,000,000,000,000 (one million billion), set it 7.94 away. Imagine what the chances of failure would be going all the way to 41.51!

Thus, the quest can now be sped up by using this information. After much discussion, I personally determined that a 1 in 10,000,000 chance of missing a record was reasonable. This is based on my personal conjecture that when a new world record is found, there are numerous other records found for iterations just below the world record. In other words, a world record has yet been found in which there exists no other records for iteration depths almost as deep as the record. Therefore, in the unlikely (1 in 10,000,000 chance) event that we will miss a record, we will likely know that we missed it, due the likelihood of numerous other numbers resolving in iteration depths very close to the limit. In this case, we can increase the iteration depth, and re-test the entire data set. Yes, it would be depressing to have to redo this work, but I believe in the 1 in 10,000,000 chance that this happens, it is worth the speed up in the program.

For a 1 in 10,000,000 chance of missing a record, our formulas determined an iteration depth of 299.63 is required, therefore I set the depth to 300 for the 18 digit set. The actual chance of missing a record for a depth of 300 iterations is 1 in 11,921,892, for those who are curious.

I would like to extend a great thanks to Vaughn Suite for his help. Using this information, the Most Delayed Palindromic Number quest has been sped up by over 5.5 times.

SUNDAY, SEPTEMBER 25, 2005 UPDATE:

On Sunday, September 25, 2005, at 3:16 am, my program (with Benjamin Despres' reversal-addition code) completed the 18-digit number set. The most delayed palindromic 18 digit number solves in 232 iterations, as you can see by the above table labelled 'Largest Delay per Digit Set'. Using this information to update the statistical analysis, we arrive at the following new formula:

Expected Maximum Iteration Depth = 14.255934 * Digit Length - 17.320261

The standard deviation was found to be 11.087996. There is a 98.96% correlation between these iteration limits and the digit length of the set they represent.

Using this information, for a 1 in 10,000,000 chance of missing a record, the iteration limit should be set to 311.20. I have rounded this value up to 315 for the 19-digit set. A limit of 315 iterations actually represents a 1 in 66,981,399 chance of missing a record.

Certain numbers do not resolve into a palindrome within the limit tested, and we believe such numbers will never solve, no matter how many iterations they are taken to. Such numbers are called Lychrel numbers. The percentage of Lychrels that occur increases with each new digit length tested:

Percentage of Lychrels
Digit Length % Not Resolved
1-digit numbers
2-digit numbers
3-digit numbers
4-digit numbers
5-digit numbers
6-digit numbers
7-digit numbers
8-digit numbers
9-digit numbers
10-digit numbers
11-digit numbers
12-digit numbers
13-digit numbers
14-digit numbers
15-digit numbers
16-digit numbers
17-digit numbers
18-digit numbers
0.00%
0.00%
1.67%
3.51%
7.25%
14.45%
22.17%
31.30%
40.42%
49.61%
57.82%
65.44%
71.64%
77.17%
81.41%
85.22%
88.03%
90.55%

Percentage Lychrels for each Digit Set

(Please note the above data is obtained from the sub-set of numbers my program tests. My program does not test every number, due to the optimizations explained earlier. So, while this graph is not 100% accurate, it is certainly very close. Most likely, it is accurate to well within the precision shown.)

CURRENT WORLD RECORD
WEDNESDAY, NOVEMBER 30, 2005 UPDATE:
Today, at 5:20 AM, my program (with Benjamin Despres' reversal-addition code) solved a new (current) world record.
The 19 digit number 1,186,060,307,891,929,990 resolves into a 119 digit palindrome in 261 iterations.
Proof: View the reversal-addition sequence for 1,186,060,307,891,929,990

This new world record beats the old world record set by my program (with Benjamin Despres' reversal-addition code) 2 years, 143 days ago, which was the 17 digit number 10,442,000,392,399,960 which resolves into a 111 digit palindrome in 236 iterations.

(I wasn't the first to beat my July 10, 2003 record of 236 iterations. A program written by Vaughn Suite found the 19-digit number 1,000,000,079,994,144,385, which resolves into a 119 digit palindrome in 259 iterations, on July 26, 2005 at 8:27 AM.)

Most Delayed Palindromic Number Records

'Zero Iterations' Note: These numbers are calculated according to the number of iterations of reversal-addition required to reach a palindrome. One iteration, at minimal, is performed. Thus, if the number is already a palindrome, it is not considered to be palindromic after 0 iterations. As a result, we have examples such as 1, 5 and 999.

'Solved' Note: In the graph below, 'solved' means that the numbers were iterated to a calculated maximum iteration needed to show within a reasonable doubt that the number will never solve.

For all 17 digit numbers and smaller, I determined it was reasonable to iterate the numbers to three times the amount of the current world record. For example, if the record is 200 iterations, my program will test other numbers to 600 iterations. This has been shown to be significant overkill in light of the statistical analysis. For the 18 digit set, Vaughn Suite and I used statistical analysis to determine a reasonable limit of 300 iterations. This limitation has a chance of less than 1 in 12,000,000 that we will miss a record. For the 19 digit set, using the same statistical analysis we determined a reasonable limit of 315 iterations. This limitation has a chance of less than 1 in 67,000,000 that we will miss a record.

It should be noted that Ian Peter's results have been essential for the assumptions made in these calculations. His extensive search performed on all 9 digit numbers and below (taking them all to 200,000 digits [slightly less than 500,000 iterations]) has shown, beyond any reasonable doubt, that if a number does not resolve into a palindrome quickly, it will never resolve into a palindrome.

Although it is believed that these numbers never solve out, like 196, it may be impossible to confirm. Until it can be proven that they will never solve, the following results are only conjectures.


Solved all 1 digit numbers
NumberIterationsDigitsResultant Palindrome
1
5
1
2
1
2
2
11
Solved all 2 digit numbers
NumberIterationsDigitsResultant Palindrome
59
69
79
89
3
4
6
24
4
4
5
13
1111
4884
44044
8813200023188
Solved all 3 digit numbers
NumberIterationsDigitsResultant Palindrome
166
188
193
829
167
849
177
999
739
989
869
187
5
7
8
10
11
14
15
16
17
19
22
23
5
6
6
8
8
10
10
10
10
11
13
13
45254
233332
233332
88555588
88555588
8836886388
8836886388
8939779398
5233333325
89540004598
8813200023188
8813200023188
Solved all 4 digit numbers
NumberIterationsDigitsResultant Palindrome
1,397
2,069
1,797
1,798
6,999
1,297
9
12
13
18
20
21
8
8
10
11
14
13
88555588
52788725
8836886388
89540004598
16668488486661
8813200023188
Solved all 5 digit numbers
NumberIterationsDigitsResultant Palindrome
10,797
10,853
10,921
10,971
13,297
10,548
13,293
17,793
20,889
80,359
13,697
10,794
15,891
70,759
70,269
10,677
10,833
10,911
25
26
27
28
29
30
31
32
33
37
38
39
40
47
52
53
54
55
16
16
16
16
18
17
17
17
17
22
22
22
22
26
28
28
28
28
1676404554046761
4455597447955544
4455597447955544
8802202552022088
893974888888479398
17858768886785871
17858768886785871
44035358885353044
44035358885353044
6839849878998789489386
6839849878998789489386
6832123695335963212386
6832123695335963212386
14525756544499444565752541
4668731596684224866951378664
4668731596684224866951378664
4668731596684224866951378664
4668731596684224866951378664
Solved all 6 digit numbers
NumberIterationsDigitsResultant Palindrome
700,269
106,977
108,933
600,259
131,996
600,279
141,996
600,579
147,996
178,992
190,890
600,589
150,296
34
35
36
45
46
50
51
57
58
59
60
63
64
22
22
22
26
26
28
28
31
31
31
31
33
33
6832123695335963212386
6832123695335963212386
6832123695335963212386
14525756544499444565752541
14525756544499444565752541
4668731596684224866951378664
4668731596684224866951378664
8834453324841674761484233544388
8834453324841674761484233544388
8834453324841674761484233544388
8834453324841674761484233544388
682049569465550121055564965940286
682049569465550121055564965940286
Solved all 7 digit numbers
NumberIterationsDigitsResultant Palindrome
1,009,227
1,007,619
1,009,246
1,008,628
1,007,377
1,001,699
1,009,150
1,058,921
1,050,995
1,003,569
1,036,974
1,490,991
3,009,179
1,008,595
1,064,912
1,998,999
7,008,429
1,000,689
1,005,744
1,007,601
7,008,899
9,008,299
41
42
43
44
48
49
56
61
62
65
66
67
68
69
70
75
77
78
79
80
82
96
23
22
23
23
28
27
28
32
33
32
32
32
35
35
35
32
38
39
39
38
38
48
68344497279697279444386
1556534287227824356551
45144454432023445444154
48852787646664678725884
8836746997299229927996476388
168977944479424974449779861
6842165664428668244665612486
18966336852467966976425863366981
682049569465550121055564965940286
14758724578598888889587542785741
14758724578598888889587542785741
14758724578598888889587542785741
46563056797844547874544879765036564
46563056797844547874544879765036564
46563056797844547874544879765036564
15521561387579888897578316512551
14674443960143265333356234106934447641
796589884324966945646549669423488985697
796589884324966945646549669423488985697
14674443960143265333356234106934447641
68586378655656964999946965655687368586
555458774083726674580862268085476627380477854555
Solved all 8 digit numbers
NumberIterationsDigitsResultant Palindrome
10,905,963
10,069,785
10,089,342
11,979,990
10,029,372
10,029,826
16,207,990
90,000,589
10,309,988
71
72
73
74
76
81
83
94
95
38
32
32
32
39
38
38
48
48
35695487976778433588533487767978459653
15521561387579888897578316512551
15521561387579888897578316512551
15521561387579888897578316512551
796589884324966945646549669423488985697
68586378655656964999946965655687368586
68586378655656964999946965655687368586
555458774083726674580862268085476627380477854555
555458774083726674580862268085476627380477854555
Solved all 9 digit numbers
NumberIterationsDigitsResultant Palindrome
100,389,898
100,055,896
110,909,992
160,009,490
800,067,199
151,033,997
100,093,573
103,249,931
107,025,910
180,005,498
100,239,862
140,669,390
84
85
86
87
88
89
90
91
92
93
97
98
45
41
40
40
46
46
46
46
46
48
52
52
584227988787668589199242991985866787889722485
88682199585544879735653797844558599128688
3525698275220268897227988620225728965253
3525698275220268897227988620225728965253
5852497685678899643696996963469988765867942585
5852497685678899643696996963469988765867942585
5852497685678899643696996963469988765867942585
5852497685678899643696996963469988765867942585
5852497685678899643696996963469988765867942585
555458774083726674580862268085476627380477854555
1345428953367763125675365555635765213677633598245431
1345428953367763125675365555635765213677633598245431
Solved all 10 digit numbers
NumberIterationsDigitsResultant Palindrome
1,090,001,921
7,007,009,909
1,009,049,407
9,000,046,899
1,050,027,948
1,304,199,693
5,020,089,949
1,005,499,526
99
100
101
103
104
105
108
109
46
49
49
49
49
49
53
53
6634544448788301675886446885761038878444454366
1543434266587555114779722279774115557856624343451
1543434266587555114779722279774115557856624343451
5831124885795990016569666669656100995975884211385
5831124885795990016569666669656100995975884211385
5831124885795990016569666669656100995975884211385
66330069478378985774345546664554347758987387496003366
66330069478378985774345546664554347758987387496003366
Solved all 11 digit numbers
NumberIterationsDigitsResultant Palindrome
10,000,505,448
10,000,922,347
10,000,696,511
10,701,592,943
10,018,999,583
10,000,442,119
10,000,761,554
10,084,899,970
10,006,198,250
18,060,009,890
11,400,245,996
16,002,897,892
18,317,699,990
37,000,488,999
10,050,289,485
90,000,626,389
10,000,853,648
13,003,696,093
10,050,859,271
10,287,799,930
10,000,973,037
10,600,713,933
10,942,399,911
60,000,180,709
11,009,599,796
16,000,097,392
10,031,199,494
10,306,095,991
10,087,799,570
102
106
107
110
111
112
113
114
115
116
117
118
119
122
123
130
131
132
133
134
135
136
137
144
145
146
147
148
149
49
50
56
53
56
56
56
56
56
56
56
56
56
58
58
65
65
65
65
65
71
71
71
66
66
66
66
66
66
5831124885795990016569666669656100995975884211385
15739929700341113302426977962420331114300792993751
13323895762789854121576789855898767512145898726759832331
23103742208899345951210026862001215954399880224730132
13656852665688105699133698688689633199650188656625865631
13656852665688105699133698688689633199650188656625865631
13656852665688105699133698688689633199650188656625865631
13656852665688105699133698688689633199650188656625865631
13656852665688105699133698688689633199650188656625865631
15997773553851169652456786055068765425696115835537779951
11144565886884377368565136533563156586377348868856544111
11144565886884377368565136533563156586377348868856544111
11144565886884377368565136533563156586377348868856544111
4618999883854436730489915000000005199840376344583889998164
4618999883854436730489915000000005199840376344583889998164
12346788678755866852059426528989698982562495025866855787688764321
12346788678755866852059426528989698982562495025866855787688764321
12346788678755866852059426528989698982562495025866855787688764321
12346788678755866852059426528989698982562495025866855787688764321
12346788678755866852059426528989698982562495025866855787688764321
12366212267356333541465465587987721212778978556456414533365376221266321
12366212267356333541465465587987721212778978556456414533365376221266321
12366212267356333541465465587987721212778978556456414533365376221266321
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
Solved all 12 digit numbers
NumberIterationsDigitsResultant Palindrome
100,900,509,906
100,000,055,859
104,000,146,950
180,005,998,298
300,000,185,539
100,001,987,765
120
121
124
129
142
143
58
58
58
65
66
66
4618999883854436730489915000000005199840376344583889998164
4618999883854436730489915000000005199840376344583889998164
4618999883854436730489915000000005199840376344583889998164
12346788678755866852059426528989698982562495025866855787688764321
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
Solved all 13 digit numbers
NumberIterationsDigitsResultant Palindrome
1,000,007,614,641
1,000,043,902,320
1,000,006,653,746
1,000,005,469,548
4,000,096,953,659
1,332,003,929,995
1,000,201,995,662
6,000,008,476,379
1,200,004,031,698
1,631,002,019,993
1,000,006,412,206
1,090,604,591,930
1,600,005,969,190
125
126
127
128
139
140
141
183
184
185
186
187
188
65
65
65
68
66
66
66
87
87
87
87
87
87
68236478976724413455368469647845654874696486355431442767987463286
68236478976724413455368469647845654874696486355431442767987463286
12346788678755866852059426528989698982562495025866855787688764321
46476268994355205755566889613764755746731698866555750255349986267464
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
895549975467412444422685224544649946445422586224444214764579945598
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
Solved all 14 digit numbers
NumberIterationsDigitsResultant Palindrome
10,090,899,969,901
40,000,004,480,279
14,104,229,999,995
138
181
182
70
87
87
2319937995996732122112578857767863993687677588752112212376995997399132
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
Solved all 15 digit numbers
NumberIterationsDigitsResultant Palindrome
100,000,109,584,608
100,000,098,743,648
100,004,789,906,151
100,079,239,995,161
100,389,619,999,030
200,000,729,975,309
107,045,067,996,994
105,420,999,199,982
101,000,269,830,970
104,000,047,066,970
700,000,001,839,569
100,000,050,469,737
101,000,789,812,993
100,907,098,999,571
100,017,449,991,820
890,000,023,937,399
100,009,989,989,199
101,507,024,989,944
107,405,139,999,943
100,057,569,996,821
103,500,369,729,970
900,000,076,152,049
100,000,439,071,028
120,000,046,510,993
103,000,015,331,997
100,617,081,999,573
100,009,029,910,821
107,000,020,928,910
100,000,090,745,299
102,000,149,322,944
130,000,074,931,591
100,120,849,299,260
150
151
152
153
154
155
156
157
158
159
163
164
165
166
167
169
170
171
172
173
174
178
179
180
189
190
191
192
198
199
200
201
80
73
80
73
73
77
77
83
77
83
78
78
77
77
77
80
76
76
82
82
80
87
87
87
87
87
87
87
92
92
92
92
14751782101782013776625853112452243335411453334225421135852667731028710128715741
3874338821379754365139707922389885874785889832297079315634579731288334783
14751782101782013776625853112452243335411453334225421135852667731028710128715741
3874338821379754365139707922389885874785889832297079315634579731288334783
3874338821379754365139707922389885874785889832297079315634579731288334783
58432004534377668812885984239766675854845857666793248958821886677343540023485
58432004534377668812885984239766675854845857666793248958821886677343540023485
66364653972239888685894387234687476484136863148467478643278349858688893227935646366
58432004534377668812885984239766675854845857666793248958821886677343540023485
66364653972239888685894387234687476484136863148467478643278349858688893227935646366
177454986735997675255356985559732595246642595237955589653552576799537689454771
177454986735997675255356985559732595246642595237955589653552576799537689454771
15951239866387982563456955855439974585458547993455855965436528978366893215951
15951239866387982563456955855439974585458547993455855965436528978366893215951
15951239866387982563456955855439974585458547993455855965436528978366893215951
36764456433123268897359635569438786398655689368783496553695379886232133465446763
1685279897987756678784477446325846776116776485236447744878766577897989725861
1685279897987756678784477446325846776116776485236447744878766577897989725861
4973663467878710242597852762388983882088668802883898832672587952420178787643663794
4973663467878710242597852762388983882088668802883898832672587952420178787643663794
36764456433123268897359635569438786398655689368783496553695379886232133465446763
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
129816518702696188228910113874486338915527747725519833684478311019822881696207815618921
129816518702696188228910113874486338915527747725519833684478311019822881696207815618921
129816518702696188228910113874486338915527747725519833684478311019822881696207815618921
129816518702696188228910113874486338915527747725519833684478311019822881696207815618921
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
Solved all 16 digit numbers
NumberIterationsDigitsResultant Palindrome
6,000,000,039,361,479
1,421,000,069,679,996
1,000,650,998,992,311
1,000,002,899,436,401
1,000,100,396,492,200
1,400,000,027,672,498
7,090,000,039,309,919
1,000,050,048,994,957
1,003,000,024,749,923
1,000,803,019,495,711
1,030,020,097,997,900
161
162
168
175
176
177
193
194
195
196
197
78
78
76
78
78
87
92
92
92
92
92
177454986735997675255356985559732595246642595237955589653552576799537689454771
177454986735997675255356985559732595246642595237955589653552576799537689454771
1685279897987756678784477446325846776116776485236447744878766577897989725861
169635646445469964587742217600000453348843354000006712247785469964544646536961
169635646445469964587742217600000453348843354000006712247785469964544646536961
159788389444247969944047982661126897487188000881784798621166289740449969742444983887951
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661
Solved all 17 digit numbers
NumberIterationsDigitsResultant Palindrome
10,000,000,730,931,027
20,005,000,862,599,819
11,450,360,479,969,994
10,009,000,275,899,569
10,059,430,139,999,234
12,179,702,595,999,991
79,000,000,445,783,599
10,000,000,767,846,797
10,000,000,673,402,336
10,000,000,525,586,206
10,005,000,760,994,249
10,030,503,899,969,524
12,000,009,694,736,291
10,442,000,392,399,960
160
206
207
208
209
210
229
230
231
232
233
234
235
236
82
90
90
90
90
90
112
112
111
111
111
111
111
111
1454578665279976634446532368777548944388998834498457778632356444366799725668754541
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
4577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754
4577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
Solved all 18 digit numbers
NumberIterationsDigitsResultant Palindrome
195,030,047,999,791,993
100,000,078,999,111,766
100,710,000,333,399,973
100,000,002,973,751,552
100,000,277,999,334,202
110,300,361,999,869,090
300,000,000,128,545,799
104,300,000,514,769,945
100,700,000,509,609,622
120,906,490,499,909,290
900,040,000,881,499,569
100,072,100,489,999,238
121,506,542,999,979,993
106,096,507,979,997,951
100,980,800,839,699,830
600,000,000,606,339,049
170,500,000,303,619,996
202
203
204
205
211
212
213
214
215
216
218
219
220
221
222
227
228
105
105
90
90
107
107
101
101
101
101
103
103
103
103
103
112
112
889956674896587435857678447634488652657823489664135444531466984328756256884436744876758534785698476659988
889956674896587435857678447634488652657823489664135444531466984328756256884436744876758534785698476659988
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
795884875720569937749728766349898266325776598895677523662898943667827947739965027578488597
16625459966833465379947700047533357850265135989944697679644998953156205875333574000774997356433866995452661
16625459966833465379947700047533357850265135989944697679644998953156205875333574000774997356433866995452661
14884695685311269599498656858759854685128966643652225634666982158645895785865689499596211358659648841
14884695685311269599498656858759854685128966643652225634666982158645895785865689499596211358659648841
14884695685311269599498656858759854685128966643652225634666982158645895785865689499596211358659648841
14884695685311269599498656858759854685128966643652225634666982158645895785865689499596211358659648841
1596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951
1596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951
1596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951
1596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951
1596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951
4577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754
4577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754
As of Saturday, January 23, 2010, solved 55.011% of all 19 digit numbers
NumberIterationsDigitsResultant Palindrome
1,000,000,038,990,407,538
1,000,000,005,577,676,468
1,060,000,000,523,124,995
1,000,000,079,994,144,385
1,003,062,289,999,939,142
1,186,060,307,891,929,990
217
225
226
259
260
261
107
111
112
119
119
119
49627604840136499793982885796699976479895651785665644844656658715659897467999669758828939799463104840672694
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975
4577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754
44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544
44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544
44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544
Iterations up to 261 for which no number was found
223, 224, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254,
255, 256, 257, 258.

The intriguing part about these records is that it takes an unreal amount of calculations to discover them, but it takes only a minute amount of calculations to prove that they terminate with a palindrome in the claimed number of iterations. With a program, it would take less than one second to prove that the above numbers do indeed solve out. I do not need a Guinness World Records Official to be present during these calculations. You can take these numbers yourself, write up a quick program, and prove the results yourself. In fact, I have done so. Click on any of the numbers above, and you will be shown a calculation page - computed in real-time (you can test this by entering your own unique number at the end of the URL) - proving that this number resolves into a palindrome in the displayed number of steps, showing you the calculations required to get there.

Please contact me if you would like more information, or if you know that I have stated incorrect information on this page. To the best of my knowledge, it is correct at this time.


FRIDAY, OCTOBER 13, 2006 UPDATE:
Vaughn Suite has completed the processing all 19 digit numbers using a distributed network of 7 machines. He has found the following records:

The 19 digit number 1,000,000,038,990,407,538 solves after 217 iterations, found on November 27, 2005.
The 19 digit number 9,000,000,000,255,353,839 solves after 224 iterations, found on February 26, 2006.
The 19 digit number 1,000,000,005,577,676,468 solves after 225 iterations, found on November 24, 2005.
The 19 digit number 1,060,000,000,523,124,995 solves after 226 iterations, found on March 9, 2006.
The 19 digit number 3,000,000,022,999,288,679 solves after 258 iterations, found on April 20, 2006.
The 19 digit number 1,000,000,079,994,144,385 solves after 259 iterations, found on November 28, 2005.
The 19 digit number 1,003,062,289,999,939,142 solves after 260 iterations, found on March 19, 2006.
The 19 digit number 1,186,060,307,891,929,990 solves after 261 iterations, found on January 2, 2006, which is the same world record that my program already solved on November 30, 2005.

No longer delays were found.

Note: You may notice that I have previously stated, in my November 30, 2005 update, that Vaughn had already found the 259 iteration record, 1,000,000,079,994,144,385, on July 26, 2005. When Vaughn was processing the 18 digit set on his work desktop last year, he started processing the 19 digit set on his laptop. This processing discovered this 259 iteration record. When he distributed the work into 15 subsets, he had to restart the 19 digit set from scratch, to be able to join it together properly. This processing rediscovered this result on November 28, 2005.

RETIREMENT:
Incidentally, at this time, I have only processed 9.782% of the 19 digit set. Why so little? My program has been running only part time, about one third of the time, on one machine. This lack of computer time is due to my retirement of this quest almost a year ago. I will eventually analyze my logs to find out when I stopped processing 24/7, and see how much CPU time I used since then until now. I should note my source code is available for any serious request to make use of it.

I retired for several reasons. One is that Vaughn is using his own implementation of the same exponentially faster algorithm that I pioneered, which means his code is as fast as mine, all else being equal.

Also, Vaughn improved the speed of the reverse and add procedure using his own assembly code (optimized for the Pentium III, Pentium 4 and Athlon XP processors). And, he did so specifically for such short iteration spans, unlike the 196 Palindrome Quest, which cannot benefit from cache as greatly, since it processes numbers much larger than available CPU cache. My reverse and add code is from Benjamin Despres, which was optimized for the 196 Palindrome Quest. As you can see from Wade's Software Comparison Page, Vaughn's code is the fastest for smaller numbers. However, even for the 196 Palindrome Quest, Vaughn's assembly code is twice as fast as Ben's.

To Ben's credit, Ben created his code a long time ago, in mid 2002, before he was able to make use of some of the newer CPU instructions. In 2002, Ben's had the fastest known application, responsible for taking the 196 Palindrome Quest from 29 million digits to 45 million digits.

Also, the largest factor in Vaughn's effort was that his application was distributed over seven machines. I only had one. I had access to seven CPUs, but they had more important things to spend their time on. Thus, I simply could not keep up with his much faster application on his incredible amount of CPU power he has available. It was a great effort, and I look forward to his analysis of the 20 digit set.

SUNDAY, FEBRUARY 3, 2008 UPDATE:
Fifteen months after completing processing the 19 digit numbers, Vaughn Suite has completed processing all 20 digit numbers by distributing the work into 15 subsets and using the Pentium 4 HT and 2 Athlon XP machines from the 19 digit processing and also three more powerful computers: an Athlon 64, an Athlon 64 X2, and a Core 2 Quad machine. The subsets were executed simultaneously on different machines, or in multiple threads on the machines with hyperthreading or multi-core capabilities (Pentium 4 HT, Athlon X2, Core-2 Quad).

Of the 5,808,378,560,022 numbers checked, there were 5,459,760,062,742 lychrels (94.00%), while 348,618,497,280 numbers resolved into palindromes.

Cumulative processing time on all the machines was 131,146,531 seconds = 1517.9 days (4 years and 2 months), but processing started January 6, 2007 and finished January 22, 2008. There are 58,083,785,600,220 (10 times as many) 21 digit numbers to be checked, so that processing of that entire set of numbers will take much longer with the current machines.

Vaughn reports that the Core-2 machine was most responsible for the quick completion since it runs optimized reverse and add software 1.7 times as fast as the next fastest processor (Athlon X2), twice as fast as the Athlon 64, 2.5 times as fast as the Athlon XPs and almost 6 times as fast as each Pentium 4 thread in Hyperthreading mode.

The new records are for 223, 253, 254, 255, 256 & 257 iterations:
The 20 digit number 10,000,000,039,513,841,287 solves after 223 iterations, found on September 20, 2007.
The 20 digit number 70,000,000,000,507,277,299 solves after 253 iterations, found on September 13, 2007.
The 20 digit number 10,200,000,000,708,183,947 solves after 254 iterations, found on March 28, 2007.
The 20 digit number 10,022,000,904,998,799,523 solves after 255 iterations, found on December 3, 2007.
The 20 digit number 10,000,000,039,395,795,416 solves after 256 iterations, found on September 20, 2007.
The 20 digit number 10,200,000,000,065,287,900 solves after 257 iterations, found on March 28, 2007.

No longer iteration delays were found, and no iterations between 237 iterations and 252 were found. The following iteration records have not yet been discovered: 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252.


Palindrome Integer Sequences

The On-Line Encyclopedia of Integer Sequences, maintained by Neil J. A. Sloane, has quite a few integer sequences regarding palindromic numbers.

My world records page is mentioned in the following:

  • A006960 - The Reverse and Add! sequence starting with 196.
  • A023108 - Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits reversed).
  • A023109 - Smallest number which requires exactly n iterations of Reverse and Add to reach a palindrome.
  • A033665 - Number of 'Reverse and Add' steps needed to reach a palindrome, or -1 if never reaches a palindrome.
  • A033670 - Reverse and Add! trajectory of 89.
  • A065198 - n sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome starting with n.
  • A065199 - Records for the number of 'Reverse and Add' steps needed to reach a palindrome.
The following three were specifically created from my Most Delayed Palindromic Number World Records:
  • A072216 - Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives values of N.
  • A072217 - Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives number of steps N takes to converge.
  • A072218 - Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives palindrome that is reached.
This On-Line Encyclopedia of Integer Sequences has over 100,000 sequences to date, and is well worth a look if you are interested in mathematical patterns.

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[Jason Doucette]: Traditional Resume / Curriculum VitaeProjects / GamesReal Time Computer GraphicsArtificial IntelligenceWorld Records / 196 Palindrome QuestWallpapers / Desktops / BackgroundsUniversity TranscriptsProgramming Windows, Fifth Edition, Errata Addendum

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