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Apply the 196-Algorithm, which consists of taking any Positive Integer of two digits or more, reversing the
digits, and adding to the original number. Now sum the two and repeat the procedure with the sum. Of the first 10,000 numbers,
only 251 do not produce a Palindromic Number in 
steps (Gardner 1979).
It was therefore conjectured that all numbers will eventually yield a Palindromic Number. However, the conjecture has been proven false for bases which are a Power of 2, and seems to be false for base 10 as well. Among the first 100,000 numbers, 5,996 numbers apparently never generate a Palindromic Number (Gruenberger 1984). The first few are 196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, ... (Sloane's A006960).
It is conjectured, but not proven, that there are an infinite number of palindromic Primes. With the exception of 11, palindromic Primes must have an Odd number of digits.
See also 196-Algorithm
References
Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American.
New York: Knopf, pp. 242-245, 1979.
Gruenberger, F. ``How to Handle Numbers with Thousands of Digits, and Why One Might Want to.''
Sci. Amer. 250, 19-26, Apr. 1984.
Sloane, N. J. A. Sequence
A006960/M5410
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.