The Palindromic Number Conjecture states that:

All integers will eventually produce a palindromic number if the 196-algorithm is repeatedly applied to them.

The conjecture has been proven false for all bases of the form 2n (base 2, base 4, base 8...), and is probably false for base 10 as well. However, it has not yet been proven true or false conclusively for base 10.
It is certainly true, however, that most numbers do produce a palindromic number. Out of the first 100,000 integers, only approximately 6,000 have not yet been shown to make a palindromic number.

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