Problem 55

Lychrel Numbers

If we take $47$ , reverse and add, $47 + 74 = 121$ , which is palindromic.

Not all numbers produce palindromes so quickly. For example,

$\begin{aligned} 349 + 943 = 1292 \\ 1292 + 2921 = 4213 \\ 4213 + 3124 = 7337 \end{aligned}$

That is, $349$ took three iterations to arrive at a palindrome.

Although no one has proved it yet, it is thought that some numbers, like $196$ , never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, $10677$ is the first number to be shown to require over fifty iterations before producing a palindrome:
$4668731596684224866951378664$ ( $53$ iterations, $28$ -digits).

Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is $4994$ .

How many Lychrel numbers are there below ten-thousand?

NOTE: Wording was modified slightly on 24 April 2007 to emphasise the theoretical nature of Lychrel numbers.

利克瑞尔数

将 $47$ 倒序并相加得到 $47 + 74 = 121$ ，是一个回文数。

不是所有的数都能像这样迅速地变成回文数。例如，

$\begin{aligned} 349 + 943 = 1292 \\ 1292 + 2921 = 4213 \\ 4213 + 3124 = 7337 \end{aligned}$

也就是说， $349$ 需要迭代三次才能变成回文数。

尽管尚未被证实，但有些数，例如 $196$ ，被认为永远不可能变成回文数。如果一个数永远不可能通过倒序并相加变成回文数，就被称为利克瑞尔数。出于理论的限制和问题的要求，我们姑且假设，对于任意一个数，除非已经通过计算证否，否则就是利克瑞尔数。此外，已知对于任意一个小于一万的数，它要么在迭代五十次以内变成回文数，要么就是没有人能够利用现今所有的计算能力将其迭代变成回文数。事实上， $10677$ 是第一个需要超过五十次迭代变成回文数的数，这个回文数是
$4668731596684224866951378664$ （ $53$ 次迭代， $28$ 位数）。

令人惊讶的是，有些回文数本身也是利克瑞尔数；第一个例子是 $4994$ 。

小于一万的数中有多少利克瑞尔数？

注意：2007年4月24日，题目略作修改，以强调目前利克瑞尔数理论的限制。