Do Lychrel numbers exist?
Take a positive whole number. Reverse its digits and add the two. Now take the
result as a new positive whole number and repeat. If, at some point, the addition results in a number of which the
digits read the same from left to right as from right to left; i.e., the number is a
palindrome, then the number that you started with is not(!) a
Lychrel number.
For example:
- 93 + 39 = 132; 132 + 231 = 363 —> 93 is not a Lychrel number.
- 1234 + 4321 = 5555 —> 1234 is not a Lychrel number.
- 157 + 751 = 908; 908 + 809 = 1717; 1717 + 7171 = 8888 —> 157 is not a Lychrel number.
- 1001 —> 1001 is a palindrome already —> 1001 is not a Lychrel number.
According to Wikipedia
"The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of "Cheryl", his girlfriend's first name."
As far as we know, no Lychrel numbers have been proven to exist, although the number 196 has been tested for
over 1 billion steps (iterations).
All positive whole numbers smaller than 196 are not Lychrel numbers.
The graph below shows for each of the numbers 1 - 195 how many iterations they required to become palindromic.
The Wikipedia page on Lychrel numbers has a list of
numbers that result in very long palindromes.
Play with this Lychrel idea below: pick a whole positive number ≤ 1000000 and the browser shows the progression of
that number until becoming a palindrome.
|