Lychrel number
A Lychrel number^{} is a number which never yields a palindrome^{} in the iterative process of adding to itself a copy of itself with digits reversed. For example, if we start with the number 983 we get:

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$983+389=1372$

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$1372+2731=4103$

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$4103+3014=7117$
So in 3 steps we get a palindrome, hence 983 is not a Lychrel number.
In fact, it is not known if there exist any Lychrel numbers in base 10 (numbers colloquially called “Lychrel numbers” in base 10 are in fact just Lychrel candidates). However, in base 2 for example, there have been numbers proven to be Lychrel numbers^{1}^{1}[2] informs us that Ronald Sprague has proved that the number 10110, for example, is a Lychrel number is base 2.. The first Lychrel candidate is 196:

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$196+691=887$

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$887+788=1675$

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$1675+5761=7436$

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$7436+6347=13783$

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$13783+38731=52514$

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$52514+41525=94039$

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$94039+93049=187088$

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$187088+880781=1067869$

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$\mathrm{\dots}$
This has been followed out to millions of digits, with no palindrome found in the sequence.
The following table gives the number of Lychrel candidates found within ascending ranges:
Range  Possible Lychrels 

0  100  0 
100  1,000  2 
1,000  10,000  3 
10,000  100,000  69 
100,000  1,000,000  99 
10,000,000  100,000,000  1728 
100,000,000  1,000,000,000  29,813 
References
 1 Wade VanLandingham, http://www.p196.org/196 And Other Lychrel Numbers
 2 John Walker, http://www.fourmilab.ch/documents/threeyears/threeyears.htmlThree Years of Computing
Title  Lychrel number 

Canonical name  LychrelNumber 
Date of creation  20130322 12:57:09 
Last modified on  20130322 12:57:09 
Owner  akrowne (2) 
Last modified by  akrowne (2) 
Numerical id  10 
Author  akrowne (2) 
Entry type  Definition 
Classification  msc 11B99 