Curious Squares

TL;DR

This paper classifies all palindromic perfect squares with a specific digit pattern, using elementary methods and elliptic curve analysis to systematically identify all such numbers.

Contribution

It provides a complete characterization of curious numbers that are perfect squares, combining elementary and advanced algebraic techniques.

Findings

Identified all curious numbers that are perfect squares.

Developed two different proof methods for the classification.

Connected the problem to the study of integral points on elliptic curves.

Abstract

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for such numbers to several single variable families. From here, we complete the proof in two different ways. The first approach is elementary, though somewhat ad hoc. The second entails studying integral points on elliptic curves and is more systematic.