A palindrome is a number that reads the same from the left side to the right side and back. For example, 1221 and 75557 are palindromes.
All natural numbers (1,2,3 and so forth to infinity) could be devided into 3 large groups:
(1) Palindromes (for example, 88, 191 or 345232543)
(2) Numbers, that become palindromes through countinuous “reverse and add” process (for example, 91: 91 + 19 = 110, 110 + 11 = 121 – palindrome)
(3) Lychrel numbers or the numbers that never-ever become palindromes regardless of steps in “reverse and add” sequence (the first and the most famous Lychrell number is 196)
The modern unresolved problems in the palindrome area are:
(1) Finding a number that requires more than 261 steps to turn into a palindrome. The official world record, established November 30, 2005, belongs to a 19-digit number (Most Delayed Palindrome World Record).
(2) Finding a sequence of unrelated (meaning not derived from each other through “reverse and add” process) numbers with a large number of iterations (over 255, or, better over 261) to turn into palindromes.
(3) Testing ranges from 20-digit numbers up to find long-iteration sequences turning initial numbers into palindromes.The brute force check of all numbers down from 20-digits took about 10 years to complete.
(4) Palindrome factorization (meaning, finding the initial sequence of seed numbers that resulted into a final palindrome) – this is a task of the future that is, similar to large integer factorization, considered “impossible to accomplish in reasonable time” for large palindromes.
We will concentrate our research efforts along all these lines.