## 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…

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

- MathematicsActa 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

- MathematicsActa Arithmetica
- 2020

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

- MathematicsQuantum 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.

## References

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- Mathematics
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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…

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…

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,…

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…

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…

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…

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…