A Partial Residue Categorization of the Magic Square of Squares

TL;DR

This paper explores the mathematical properties and residue classifications of hypothetical 3x3 magic squares composed of squared integers, providing a framework for understanding their possible structures and conjecturing about their existence.

Contribution

It introduces a residue categorization method for the magic square of squares based on prime divisors of the central entry, linking to finite field solutions and proposing new conjectures.

Findings

Residue categorization of magic squares of squares based on prime divisors

Connection between residue classes and finite field solutions

Conjectures on the existence of certain residue configurations

Abstract

It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same total. We give a categorization of the residue of the magic square of squares for primes dividing the central entry. The categorization leads to a magic square of square solution over a finite field and, lastly, some conjectures are made regarding the existence of certain residues.