# 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} }

Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the… Expand

#### 21 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… Expand

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… Expand

Sums of four squares with certain restrictions.

- Mathematics
- 2020

Let $a,b\in\mathbb N=\{0,1,2,\ldots\}$ and $\lambda\in\{2,3\}$. We show that $4^a(4b+1)$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $x+2y+\lambda z$ is a positive… Expand

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… Expand

Extended Lagrange's four-square theorem

- Computer Science, Mathematics
- Electron. Notes Discret. Math.
- 2018

This article proves that any system of orthogonal vectors of norm p can be completed to a base in Z^4 and conjecture that the result holds for every norm $p\geq 1$. Expand

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… Expand

On the 1-3-5 conjecture and related topics

- Mathematics
- 2017

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… Expand

Sums of integers and sums of their squares

- Mathematics
- 2019

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… Expand

N T ] 3 J un 2 01 9 SUMS OF SQUARES WITH RESTRICTIONS INVOLVING PRIMES

- 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… Expand

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 $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$… Expand

#### References

SHOWING 1-10 OF 13 REFERENCES

A result similar to Lagrange's theorem

- Mathematics
- 2015

Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers… Expand

On almost universal mixed sums of squares and triangular numbers

- Mathematics
- 2008

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… Expand

Sums of Squares of Integers

- Mathematics
- 2005

Introduction Prerequisites Outline of Chapters 2 - 8 Elementary Methods Introduction Some Lemmas Two Fundamental Identities Euler's Recurrence for Sigma(n) More Identities Sums of Two Squares Sums of… Expand

On universal sums of polygonal numbers

- Mathematics
- 2009

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,… Expand

Ramanujan's ternary quadratic form

- Mathematics
- 1997

do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from… Expand

Integers represented by positive ternary quadratic forms

- Mathematics
- 1927

Without giving any details, he stated that like considerations applied to the representation of multiples of 3 by B. But the latter problem is much more difficult and no treatment has since been… Expand

The representation of integers as sums of squares

- Mathematics
- 2002

<abstract abstract-type="TeX"><p>We present a uniform method by which we obtain an explicit formula for the number of representations of an integer as the sum of <i>n</i> squares for each <i>n</i> in… Expand

Additive Number Theory The Classical Bases

- Mathematics
- 1996

The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial… Expand