Quotients of Palindromic and Antipalindromic Numbers

TL;DR

This paper investigates the properties of quotients formed by binary palindromic and antipalindromic numbers, establishing decidability results, bounds, and density properties of representable integers.

Contribution

It introduces the first algorithmic decision procedure for representing integers as quotients of palindromic numbers and analyzes the density of unrepresentable integers.

Findings

Decidability of representing an integer as a quotient of palindromic numbers

Bound on the numerator size in minimal representations

Positive density of unrepresentable integers

Abstract

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.