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。