Lychrel Number - Proof Not Found

Proof Not Found

In other bases, certain numbers can be proven to never form a palindrome after repeated reversal and addition, but no such proof has been found for 196 and other base 10 numbers.

It is conjectured that 196 and other numbers which have not yet yielded a palindrome are Lychrel numbers, but no number has yet been proven to be Lychrel. Numbers which have not been demonstrated to be non-Lychrel are informally called "candidate Lychrel" numbers. The first few candidate Lychrel numbers (sequence A023108 in OEIS) are:

196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997.

The numbers in bold are suspected Lychrel seed numbers (see below). Computer programs by Jason Doucette, Ian Peters and Benjamin Despres have found other Lychrel candidates. Indeed, Benjamin Despres' program has identified all suspected Lychrel seed numbers of less than 17 digits. Wade VanLandingham's site lists the total number of found suspected Lychrel seed numbers for each digit length.

The brute-force method originally deployed by John Walker has been refined to take advantage of iteration behaviours. For example, Vaughn Suite devised a program that only saves the first and last few digits of each iteration, enabling testing of the digit patterns in millions of iterations to be performed without having to save each entire iteration to a file. But so far no algorithm has been developed to circumvent the reversal and addition iterative process.