On partitions of integers with restrictions involving squares

TL;DR

This paper proves two conjectures about representing integers as sums with restrictions involving squares, identifying specific exceptions and providing new insights into such partitions.

Contribution

It establishes two conjectured results on integer partitions with square-related restrictions, including exceptions, advancing understanding in this area.

Findings

Most positive integers can be expressed as sums with square restrictions, except specific forms.

Certain integers greater than 7 can be expressed as sums with scaled square restrictions.

The paper confirms conjectures posed by Sun in 2013.

Abstract

In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013): (i) Each positive integer $n$ can be written as $n=x+y+z$ with $x,y,z$ positive integers such that $x^2+y^2+z^2$ is a square, unless $n$ has the form $n=2^{a}3^{b}$ or $2^{a}7$ with $a$ and $b$ nonnegative integers. (ii) Each integer $n>7$ with $n\not=11,14,17$ can be written as $n=x+y+2z$ with $x,y,z$ positive integers such that $x^2+y^2+2z^2$ is a square.