## Magic Square of Squares - The Modular Approach

**Goal:**

A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same.
In previous projects in the Experimental Mathematics Lab, students have created very interesting and beautiful "magic objects".

A magic square of squares is a magic square such that each of its entries is a square.
It is an open problem to decide if a 3x3 magic square of squares exists. For the 4x4 case, Euler gave a solution, which in modern algebra terms stems from the multiplicativity of the norm of quaternions.

Many variations are possible, for instance, magic tetrahedra of squares. Using some linear algebra combined with a "brute-force" search, I produced an example.

The goal this term is to push the "modular method" to obtain "magic objects of squares" or to compute lower bounds for the size of the entries in potential magic objects of squares.
More precisely, the "modular method" will produce necessary conditions that entries in a magic object need to satisfy (usually in forms of congruences). One can then run through all tuples of integers (up to a bound given by the available computing time) satisfying these necessary conditions and either produce magic objects of squares or prove that any magic object of squares must have entries greater than the bound of the computations.

**Supervisors:**
Thierry Meyrath, Gabor Wiese

**Difficulty level:**
Any.

**Tools:**

- Any programming language or any Computer Algebra System such as sagemath
- Image processing software

**Literature:**

- Christian Boyer: Some Notes on the Magic Squares of Squares Problem, The Mathematical Intelligencer, 2005
- O. M. Cain: Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236