Are numbers that are already palindromes Lychrel numbers?

Question

I'm working on a Project Euler problem where I'm supposed to determine Lychrel numbers. I'm not sure I understand what a Lychrel number is.

From my understanding a Lychrel number is any number that is a palindrome or that can become a palindrome by adding the reverse of a number to said number.

Examples

Digits $1$ to $9$ => Because any 1 digit number is a palindrome

$121$ => Reads the same way left to right

$47$ => $47 + 74$ is $121$, which is a palindrome

So my question is are numbers that are already palindromes Lychrel numbers, or do they need to be added to another number to become Lychrel numbers?

Wikipedia Article

http://en.wikipedia.org/wiki/Lychrel_number

Edit: Looks like I got the definition mixed up. A Lychrel number is actually a number that can never become a palindrome. I apologize for any confusion I caused.

Answer 1

My knee-jerk response is: no, if it's a palindrome, then it's not a Lychrel number. But your question deserves more than a knee-jerk response.

I suppose there is no way to guard against getting confused (happens to the best of us), but at least you should avoid that source of information famous for slandering a respected journalist. In your Google search for "Lychrel number", Wikipedia was your first result, but Mathworld was probably your second result. Ignore Wikipedia and look at what Mathworld says:

The first few numbers not known to produce palindromes when applying the 196-algorithm (i.e., a reverse-then-add sequence) are sometimes known as Lychrel numbers. ... The first few Lychrel numbers are 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, ... (OEIS A023108).

But there is still room for argument here. One could argue that we're looking to see if a palindrome is "produced," meaning that at least one reverse-then-add step must be performed regardless of whether the number is a palindrome to begin with or not. So 121 is already a palindrome, but if we take it through the reverse-then-add, we get:

$121 + 121 = 242$

and that is a palindrome, so 121 is not a Lychrel number because it produces a palindrome right after the first reverse-then-add. What about 242? Same thing. Things get more interesting for 484:

$484 + 484 = 968$

$968 + 869 = 1837$

$1837 + 7381 = 9218$

$9218 + 8129 = 17347$

$17347 + 74371 = 91718$

This has already been computed to many digits, see Sloane's A033650. But either way you slice it, 484 is not a Lychrel number. If we insist it must go through at least one reverse-then-add step, then, after about a couple dozen iterations, we get 8813200023188.