On Polynomial Pairs of Integers

TL;DR

This paper explores polynomial pairs of integers, showing they are always palindromic and conjecturing that all non-palindromic integers forming palindromic pairs are polynomial, linking to classical recreational math topics.

Contribution

It introduces polynomial pairs, proves they are always palindromic, and conjectures their equivalence to all non-palindromic palindromic pairs, connecting to known recreational math concepts.

Findings

Polynomial pairs are always palindromic.

Conjecture: all non-palindromic palindromic pairs are polynomial.

Connections to reversal multiplication, palindromic squares, and repunits.

Abstract

The reversal of a positive integer $A$ is the number obtained by reading $A$ backwards in its decimal representation. A pair $(A,B)$ of positive integers is said to be palindromic if the reversal of the product $A \times B$ is equal to the product of the reversals of $A$ and of $B$. A pair $(A,B)$ of positive integers is said to be polynomial if the product $A \times B$ can be performed without carry. In this paper, we use polynomial pairs in constructing and in studying the properties of palindromic pairs. It is shown that polynomial pairs are always palindromic. It is further conjectured that, provided that neither $A$ nor $B$ is itself a palindrome, all palindromic pairs are polynomial. A connection is made with classical topics in recreational mathematics such as reversal multiplication, palindromic squares, and repunits.