On Integers Whose Sum is the Reverse of their Product

TL;DR

This paper characterizes all positive integer pairs where the sum's digits are the reverse of the product's digits, using automata theory, and extends the approach to various numerical bases.

Contribution

It introduces a novel automata-based method to identify such integer pairs and generalizes the concept to different numerical bases.

Findings

Identified all such integer pairs in base 10.

Developed automata that describe the problem's solutions.

Extended the approach to other numerical bases.

Abstract

We determine all pairs of positive integers $(a,b)$ such that $a+b$ and $a \times b$ have the same decimal digits in reverse order: \[ (2,2), (9,9), (3,24), (2,47), (2,497), (2,4997), (2,49997), \ldots \] We use deterministic finite automata to describe our approach, which naturally extends to all other numerical bases. Our automata are a variation on the notion of Young graphs, which were introduced by Sloane to study ``reverse multiples''.