论文标题
四位数Kaprekar流程中的最大距离
Maximum distances in the four-digit Kaprekar process
论文作者
论文摘要
对于自然数字$ x $和$ b $，经典的kaprekar函数定义为$ k_ {b}（x）= d-a $，其中$ d $是基本的重新安排-B $ b $ b $ b $ b $ a $ x $的降序，$ a $ a $是上升。 $ k_b $的基础$ b $由Hasse和Prichett分类为$ 4 $ digit的非零固定点，而对于每个基础，该固定点是唯一的。在本文中，我们确定在所有四位数基础中达到此固定点所需的最大迭代次数-B $输入，从而回答了Yamagami的问题。此外，我们还探索---作为$ b $ ---四位数输入的函数，其迭代$ k_b $收敛到该固定点。
For natural numbers $x$ and $b$, the classical Kaprekar function is defined as $K_{b} (x) = D-A$, where $D$ is the rearrangement of the base-$b$ digits of $x$ in descending order and $A$ is ascending. The bases $b$ for which $K_b$ has a $4$-digit non-zero fixed point were classified by Hasse and Prichett, and for each base this fixed point is known to be unique. In this article, we determine the maximum number of iterations required to reach this fixed point among all four-digit base-$b$ inputs, thus answering a question of Yamagami. Moreover, we also explore---as a function of $b$---the fraction of four-digit inputs for which iterating $K_b$ converges to this fixed point.
