A Constraint Propagation Algorithm for Sums-of-Squares Formulas over the Integers

TL;DR

This paper introduces an algorithm that constructs or proves the nonexistence of sums-of-squares formulas over integers by generating signed intercalate matrices, extending previous computational methods.

Contribution

It presents a novel algorithm that automates the search for signed intercalate matrices, enabling the study of sums-of-squares formulas beyond manual computational limits.

Findings

Algorithm successfully constructs signed intercalate matrices.

Proves nonexistence of certain sums-of-squares formulas.

Extends computational feasibility for this class of problems.

Abstract

Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce sums-of-squares formulas and to show that formulas of certain types cannot exist. In this paper, we introduce an algorithm that produces consistently signed intercalate matrices, or proves their nonexistence, extending previous methods beyond what is computationally feasible by hand.