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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 numbers 1. 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 |
- 1
- Wade VanLandingham, 196 And Other Lychrel Numbers
- 2
- John Walker, Three Years of Computing
Footnotes
- 1
- [2] informs us that Ronald Sprague has proved that the number 10110, for example, is a Lychrel number is base 2.
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