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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{align} 349 + 943 &= 1292\\ 1292 + 2921 &= 4213\\ 4213 + 3124 &= 7337 \end{align}

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.

Lychrel数¶

47とその反転を足し合わせると, 47 + 74 = 121となり, 回文数になる.

全ての数が素早く回文数になるわけではない. 349を考えよう,

1. 349 + 943 = 1292,
2. 1292 + 2921 = 4213
3. 4213 + 3124 = 7337

349は, 3回の操作を経て回文数になる.

まだ証明はされていないが, 196のようないくつかの数字は回文数にならないと考えられている. 反転したものを足すという操作を経ても回文数にならないものをLychrel数と呼ぶ. 先のような数の理論的な性質により, またこの問題の目的のために, Lychrel数で無いと証明されていない数はLychrel数だと仮定する.

更に, 10000未満の数については，常に以下のどちらか一方が成り立つと仮定してよい.

1. 50回未満の操作で回文数になる
2. まだ誰も回文数まで到達していない

実際, 10677が50回以上の操作を必要とする最初の数である: 4668731596684224866951378664 (53回の操作で28桁のこの回文数になる).

驚くべきことに, 回文数かつLychrel数であるものが存在する. 最初の数は4994である.

10000未満のLychrel数の個数を答えよ.

注: 2007/04/24にLychrel数の理論的な性質を強調するために文面が修正された.