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:

•

$983+389=1372$
•

$1372+2731=4103$
•

$4103+3014=7117$

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 numbers¹¹[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:

•

$196+691=887$
•

$887+788=1675$
•

$1675+5761=7436$
•

$7436+6347=13783$
•

$13783+38731=52514$
•

$52514+41525=94039$
•

$94039+93049=187088$
•

$187088+880781=1067869$
•

$\ldots$

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	2013-03-22 12:57:09
Last modified on	2013-03-22 12:57:09
Owner	akrowne (2)
Last modified by	akrowne (2)
Numerical id	10
Author	akrowne (2)
Entry type	Definition
Classification	msc 11B99