the wikipedia page explains it fairly well https://en.wikipedia.org/wiki/Lychrel_number

its a number where reversing it and adding it to its self repeatedly never becomes a palindrome (symetric string). for example 437, reverse it, and you get 734, add 437 and you get 1171, repeat to get 1711+1171= 2882, a palindrome 28|82, so it is not a Lychrel number

196 (first 4 itterations are 887, 1675, 7436, 13783) might be, but none have actually been proved.

as for a use, who knows. we dont know if any actually exist. I would guess there is no use, but mathematics has a habit of making up “useless” concepts that turn out to be unexpectedly useful 100 years later

Take any natural number, lets use 175 for today. Write it in reverse: 571. Now add the two to get 175 + 571 = 746.

Okay, then do the same again, reverse and add: 746 + 647 = 1393.

And again: 1393 + 3931 = 5334.

Aaaaand one more time: 5334 + 4335 = 9669.

We now arrived at a _palindrome_, a number that looks the same if we reverse it. Does this always happen, regardless of what number we started with? It might take four or a hundred or a bazillion steps for some number to finally settle; but do all of them eventually do so? The answer is: we don’t know. So we decided to call those hypothetical numbers where this process never spits out a palindrome “Lychrel numbers”.

So we have no idea if they exist. Apart from some low amount of fame, this question is pretty inconsequential, though. It has absolutely no suspected relation to real life at all. It isn’t even important for almost anything else within mathematics.

It nonetheless is difficult problem and in trying or succeeding to solve it, we might learn new methods or grasp concepts that could help with more important things. Or we just try because we find it interesting in itself.

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