For the last year or so I've been thinking about what seems to be a fairly obscurenumber-theoretic problem. I'll do my best to describe it here.
This problem is all about the reverse-and-add algorithm. I.e., you reverse an integer's digits (in a certain base)and add it to itself over and over. The sequence, if it terminates, ends when the digits of the sum are palindromic. Palindromic numbers are the same forwards and backwards, for example: 696, 133312213331, 33, 121, 0xBEEFEEB, etc.
Here's what that looks like for the number 48(in base 10):
48
+84
------
132
+231
------
363
A palindrome was reachedin two additions. The most number additions required by any base-10 integer less than 10,000 is 89, which requires 24 iterations to become 8813200023188(a palindrome).
The majority of integers below about 1018do eventually become palindromes, however some of them seem toneverbecome palindromic. These are called Lychrel numbers. It's conjectured that 196 is the lowest Lychrel number in base-10, however there is no proof that base-10 Lychrel numbers exist. Lychrel numbers are proven to exist in binary and a few other bases however.
Higher integers than 1018seem to become more likely to be Lychrel.
The most-delayed palindromic number, or the most delayed non-Lychrel number, is the number that takes the most steps to eventually become a palindrome under the reverse-and-add algorithm.
In a future post I'll describe my efforts to beat the current most-delayed palindromic numberat 261 steps.
