A Lychrel number is a natural number that cannot form a palindrome by the 196-algorithm: an iterative process of repeatedly reversing a numbers’ digits and adding the resulting numbers.
Whilst in other bases (powers of two) certain numbers can be proven to never form a palindrome, in base 10 (the base system we use in everyday life) no Lychrel numbers have been proven to exist. However many numbers, such as 196, are suspected to be a Lychrel number on “heuristic and statistical grounds“.
The name Lychrel was coined by Wade Van Landingham in 2002 as an anagram of Cheryl, his girlfriend’s name.
The reverse-and-add process is when you add a number to the number formed by reversing the order of its digits.
Examples of non-Lychrel numbers are (taken from Wikipedia):
- 56 becomes palindromic after one iteration: 56+65 = 121.
- 57 becomes palindromic after two iterations: 57+75 = 132, 132+231 = 363.
- 59 becomes a palindrome after 3 iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111
- 89 takes an unusually large 24 iterations to reach the palindrome 8,813,200,023,188.
- 1,186,060,307,891,929,990 takes 261 iterations to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which is the current world record for the Most Delayed Palindromic Number. It was solved by Jason Doucette‘s algorithm and program in 2005.
196 is the smallest number suspected to never reach a palindrome in base 10 and has thus received the most attention:
- In 1985 a program by James Killman ran unsuccessfully for over 28 days, cycling through 12,954 passes and reaching a 5366-digit number.
John Walker began his 196 Palindrome Quest in 1987. His program ran for almost three years, then terminated (as instructed) in 1990 with the message:
- Stop point reached on pass 2,415,836.
- Number contains 1,000,000 digits.
- In 1995, Tim Irvin and Larry Simkins reached the two million digit mark in only three months without finding a palindrome.
- Jason Doucette then reached 12.5 million digits in May 2000.
- Wade Van Landingham used Jason Doucette’s program to reach a 13 million digit. By May 2006, Van Landingham had reached the 300 million digit mark.
- In 2011 Romain Dolbeau completed a billion iterations to produce a number with 413,930,770 digits, and in February 2015 his calculations reached a number with billion digits.
A palindrome has yet to be found.