回文数和反回文数的商
Quotients of Palindromic and Antipalindromic Numbers
摘要：
一个自然数N被称为回文数，如果它的二进制表示读起来是相同的向前和向后。本文研究了两个回文数的乘积，并回答了关于整数和有理数的结果集的一些基本问题。例如，我们证明了以下问题在算法上是可判定的：给定一个整数N，确定我们是否可以对回文数A和B写出N = A/B。假设N是可表示的，我们找到最小表示的分子大小的界。我们证明了不可表示整数集在N中具有正密度。我们也得到了类似的结果，反回文数（那些二进制表示的前半部分是后半部分的逆补数）。我们还提供了例子，数值数据，和一些有趣的插图和开放的问题。
英文摘要：
A natural number N is said to be palindromic if its binary representation reads the same forwards and backwards. In this paper we study the quotients of two palindromic numbers and answer some basic questions about the resulting sets of integers and rational numbers. For example, we show that the following problem is algorithmically decidable: given an integer N, determine if we can write N = A/B for palindromic numbers A and B. Given that N is representable, we find a bound on the size of the numerator of the smallest representation. We prove that the set of unrepresentable integers has positive density in N. We also obtain similar results for quotients of antipalindromic numbers (those for which the first half of the binary representation is the reverse complement of the second half). We also provide examples, numerical data, and a number of intriguing conjectures and open problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
