Search for the biggest numeric palindrome

Page last updated -- Thursday August 19 1999

What it's all about

The following description has been shamelessly stolen from John Walker's palindrome quest page

Pick a number. Reverse its digits and add the resulting number to the original number. If the result isn't a palindrome, repeat the process. Do all numbers in base 10 eventually become palindromes through this process? Nobody knows. For example, start with 87. Applying this process, we obtain: 

    87 + 78 = 165
              165 + 561 = 726
                          726 + 627 = 1353
                                      1353 + 3531 = 4884, a palindrome

Whether all numbers eventually become palindromic under this process is unproved, but all numbers less than 10,000 have been tested. Every one becomes a palindrome in a relatively small number of steps (of the 900 3-digit numbers, 90 are palindromes to start with and 735 of the remainder take less than 5 reversals and additions to yield a palindrome). Except, that is, for 196. This number had been carried through 50,000 reversals and additions by P. C. Leyland, yielding a number of more than 26,000 digits without producing a palindrome. Later, P. Anderton continued the process up to 70,928 digits without encountering a palindrome.

John then goes on to explain how, over the three years prior to 1990, he took this number out to 1,000,000 (one million) digits without it becoming a palindrome. Others have taken it further.

This project deals with a different aspect of these palindromic numbers.

If you start with the number 10911, reverse, add etc. until it becomes a palindrome, it will take you 55 'goes', or iterations, and produce the number 

          4668 7315 9668 4224 8669 5137 8664
which has 24 digits and is a palindrome.

This was the most 'delayed' palindrome know to the author in 1990/1991. (There are, of course, many numbers that converge on the same 24 digit palindrome - trivially 11901.)

There are more 'base' numbers that do not seem to form palindromes and do not seem to converge onto the same sequence. The first six are 196, 879, 1997, 7059, 10553 and 10563.

How many are there?

At this stage a cut off point of 1000 digits was chosen and a search between 1 and 9,999,999 was started.

It was found that there are 1895 such numbers.

The big project

Looking for the most 'delayed' palindrome.

In 1991, calculations showed that it was possible to test all 1895 numbers up to a length of 200,000 digits in a realistic time frame (measured in years).

The following are the machines that ran the program at various times at the author's home.

Other people also helped and have run the program on the following configurations.

The results

In may 1996 the final number (9907699) was reached. It is esimated that 90% of the numbers were checked in less than the final 50% of the time. (This was, of course, expected.)

The following are the numbers, in the order they were found, that take the greatest number of iterations to form a palindome. 

 Start number  Iterations     Palindrome formed

    147996        58    8834453324841674761484233544388
    150296        64    682049569465550121055564965940286
    1000689       78    796589884324966945646549669423488985697
    1005744       79    796589884324966945646549669423488985697
    1017501       80    14674443960143265333356234106934447641
    7008899       82    68586378655656964999946965655687368586
    9008299       96    555458774083726674580862268085476627380477854555

This shows that, there is no base number between 1 and 9,999,999 that forms a palindrome in greater than 96 iterations up to a length of 200,000 digits ( slightly less than 500,000 iterations - variable).

The latest results

Work continues on the the numbers between 10,000,000 and 1,999,999,999. So far, the results show 

 Start number  Iterations     Palindrome formed
    100239862     97      1345428953367763125675365555635765213677633598245431
    140669390     98      1345428953367763125675365555635765213677633598245431
   1090001921     99      6634544448788301675886446885761038878444454366
   1009049407    101      1543434266587555114779722279774115557856624343451
   1050027948    104      5831124885795990016569666669656100995975884211385
   1304199693    105      5831124885795990016569666669656100995975884211385
   1005499526    109      66330069478378985774345546664554347758987387496003366
[Note the 'out of order' start numbers]

The search hasn't stopped. More info will follow...

You can go to my home page or...

Send mail to ijp@floot.demon.co.uk if you have, or want, more information.