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
View PDF on arXiv
22 Citations
Highly Influential Citations
1
Background Citations
14
Methods Citations
5

22 Citations

Some refinements of Lagrange's four-square theorem
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
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
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
Sums of squares with restrictions involving primes
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
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
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
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
TLDR
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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References

SHOWING 1-10 OF 13 REFERENCES
A result similar to Lagrange's theorem
On almost universal mixed sums of squares and triangular numbers
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
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
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
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
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
<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
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…
Quaternary Quadratic Forms Representing all Integers
Modern Elementary Theory of Numbers
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