It is true that technically speaking there are no PROVEN Lychrel numbers. There have been no mathematical proofs saying that all numbers will become palindromes, and none that say certain ones never will...
HOWEVER...Some people have spent a LOT of money working on these numbers (especially 196)...they have gone through millions of iterations and taken the number out to over 300 million digits and still no palindrome. And when you stop to think about it probabilistically, the larger the number becomes the less likely it will ever become a palindrome. There are plenty of websites out there that are "devoted" to Lychrel numbers and the search for palindromes...so, you could just do a search for Lychrel 196 or something and find these sites.
I don't know who thought of this whole process...but Wade VanLandingham came up with the Lychrel numbers (or at least the name for them)...
I wrote a simple program to determine which numbers become palindromes...and so, that's probably how someone figured out that most of them were NOT Lychrel numbers.
Hope this provides some insight.