Zhi-Wei Sun's 1-3-5 conjecture and variations

被引：6

作者：

Machiavelo, Antonio ^{[1
]}

Tsopanidis, Nikolaos ^{[1
]}

机构：

[1] Univ Porto, Ctr Matemat, P-4169007 Porto, Portugal

来源：

JOURNAL OF NUMBER THEORY | 2021年 / 222卷

关键词：

Quaternions; Lipschitz integers; 1-3-5; conjecture;

D O I：

10.1016/j.jnt.2020.10.001

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhi-Wei Sun's "1-3-5 conjecture" for all integers, and reduce the general case to its verification up to 1.052 x 10(11). The computational verification was performed by the authors and a colleague, concluding the proof of Sun's 1-3-5 conjecture. We also establish some variations of this conjecture. (C) 2020 Elsevier Inc. All rights reserved.

引用

页码：1 / 20

页数：20

共 11 条

[1]

Ankeny N., 1957, Proceedings of the American Mathematical Society, V8, P316

[2]

[Anonymous], 2003, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry

[3]

Legendre AM., 1808, ESSAI THEORIE NOMBRE

[4]

Lehmer D.H., 1948, Am. Math. Mon., V55, P476

[5] On the arithmetic of quaternions [J].

Pall, Gordon .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1940, 47 (1-3) :487-500

[6] Some variants of Lagrange's four squares theorem [J].

Sun, Yu-Chen ;

Sun, Zhi-Wei .

ACTA ARITHMETICA, 2018, 183 (04) :339-356

[7] Restricted sums of four squares [J].

Sun, Zhi-Wei .

INTERNATIONAL JOURNAL OF NUMBER THEORY, 2019, 15 (09) :1863-1893

[8] Refining Lagrange's four-square theorem [J].

Sun, Zhi-Wei .

JOURNAL OF NUMBER THEORY, 2017, 175 :167-190

[9]

Tsopanidis N, 2020, ARXIV200513526

[10]

Wojcik J., 1971, Colloquium Mathematicum, V1, P117

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