Lagrange's Theorem On The Minimal Set Of Squares

TL;DR

This paper demonstrates the existence of a minimal thin basis of order four with cardinality growing as O(x^0.25), advancing understanding of minimal bases in number theory.

Contribution

It establishes the existence of a thin basis of order four with minimal cardinality, improving upon previous bounds and confirming conjectures about minimal bases.

Findings

Existence of a thin basis of order four with cardinality O(x^0.25)

Improves bounds on minimal bases in additive number theory

Supports conjectures on minimal thin bases

Abstract

It will be demonstrated that there is a thin basis of order four of minimal cardinality #A(x) = O(x^.25). The current literature shows the existence of a thin basis of order four of cardinality #A(x) = O(x^(.25+{\epsilon}), {\epsilon} > 0, but speculates on the existence of a thin basis of order four of minimal cardinality.