A result similar to Lagrange's theorem

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• 作者：Zhi-Wei Sun ^{zwsun@nju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11E25 ; secondary ; 11B75 ; 11D85 ; 11E20
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：162
• 期：Complete
• 页码：190-211
• 全文大小：413 K

文摘

Generalized octagonal numbers are those dd1929645c46c0eeef210f7f" title="Click to view the MathML source">p₈(x)=x(3x−2)$p_8(x)=x(3x-2)$ with x∈Z$x\in \mathbb{Z}$. In this paper we show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for 35 triples (b,c,d)$(b,c,d)$ with cdd888a8f224afc297db26c02e0" title="Click to view the MathML source">1⩽b⩽c⩽d$1⩽b⩽c⩽d$ (including (2,3,4)$(2,3,4)$ and dd1dbf08bcd0d637981f7a26fb" title="Click to view the MathML source">(2,4,8)$(2,4,8)$), we prove that any nonnegative integer can be expressed as p₈(w)+bp₈(x)+cp₈(y)+dp₈(z)$p_8(w)+b{p}_8(x)+c{p}_8(y)+d{p}_8(z)$ with w,x,y,z∈Z$w,x,y,z\in \mathbb{Z}$. We also pose several conjectures for further research.