# A parametrization of 8x8 magic squares of squares through octonionic multiplication

## TL;DR

This paper explores a novel parametrization method for 8x8 magic squares of squares with orthogonal rows using octonionic multiplication, extending quaternionic techniques but highlighting limitations in further Cayley-Dickson extensions.

## Contribution

It introduces a new octonionic approach to parametrizing 8x8 magic squares of squares, building on Euler's quaternionic methods and identifying the boundaries of Cayley-Dickson extensions.

## Key findings

- Parametrization achieved via octonions for 8x8 magic squares of squares.
- Extension to higher Cayley-Dickson algebras is not feasible.
- Method parallels Euler's quaternionic construction for 4x4 matrices.

## Abstract

In an analogous construction as by Euler for 4x4 matrices, a parametrization of 8x8 magic squares of squares with orthogonal rows is shown to be obtainable by extending the quaternionic method, as shown by Hurwitz, to octonions, but not possible to be carried even further in the Cayley-Dickson construction series.