# 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 LychrelNumber 2013-03-22 12:57:09 2013-03-22 12:57:09 akrowne (2) akrowne (2) 10 akrowne (2) Definition msc 11B99