# Refining Lagrange's four-square theorem

@article{Sun2016RefiningLF,
title={Refining Lagrange's four-square theorem},
author={Zhi-Wei Sun},
journal={arXiv: Number Theory},
year={2016}
}
• Zhi-Wei Sun
• Published 22 April 2016
• Mathematics
• arXiv: Number Theory
22 Citations
Some refinements of Lagrange's four-square theorem
• Mathematics
• 2016
Lagrange's four-square theorem is a classical theorem in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and
Some variants of Lagrange's four squares theorem
• Mathematics
• 2016
Lagrange's four squares theorem is a classical theorem in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and
Further results on sums of squares
• Mathematics
• 2018
The well-known Lagrange's four-square theorem states that any integer $n\in\mathbb{N}=\{0,1,2,...\}$ can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of
Extended Lagrange's four-square theorem
• Mathematics
Electron. Notes Discret. Math.
• 2018
Sums of squares with restrictions involving primes
• Mathematics
• 2018
The well-known Lagrange's four-square theorem states that any integer $n\in\mathbb{N}=\{0,1,2,...\}$ can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of
On the 1-3-5 conjecture and related topics
• Mathematics
Acta Arithmetica
• 2020
The 1-3-5 conjecture of Z.-W. Sun states that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x^2+y^2+z^2+w^2$ with $w,x,y,z\in\mathbb N$ such that $x+3y+5z$ is a square. In this paper, via
Sums of integers and sums of their squares
Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers
N T ] 3 J un 2 01 9 SUMS OF SQUARES WITH RESTRICTIONS INVOLVING PRIMES
• Mathematics
• 2019
The well-known Lagrange’s four-square theorem states that any integer n ∈ N = {0, 1, 2, ...} can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of n as x
Proof of a conjecture of Sun on sums of four squares
• Mathematics
• 2020
In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$
Discrete quantum computation and Lagrange's four-square theorem
• Mathematics
Quantum Inf. Process.
• 2020
The simplest version of the problem, 2-qubit systems with integers with integers as coordinates, but with normalization factor p, proves that any system of orthogonal vectors of norm p can be completed to a basis.

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In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous
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For m = 3, 4, …, the polygonal numbers of order m are given by pm(n) = (m − 2)(2n) + n (n = 0, 1, 2, …). For positive integers a, b, c and i, j, k ⩾ 3 with max{i, j, k} ⩾ 5, we call the triple (api,