On x(ax + 1)+y(by + 1)+z(cz + 1) and x(ax + b)+y(ay + c)+z(az + d)

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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 ; 11D85 ; 11E20
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：275-283
• 全文大小：274 K

文摘

In this paper we first investigate for what positive integers a,b,c$a,b,c$ every nonnegative integer n   can be written as x(ax+1)+y(by+1)+z(cz+1)$x(ax+1)+y(by+1)+z(cz+1)$ with x,y,z$x,y,z$ integers. We show that (a,b,c)$(a,b,c)$ can be either of the following seven triples and conjecture that any triple (a,b,c)$(a,b,c)$ among also has the desired property. For integers 0⩽b⩽c⩽d⩽a$0⩽b⩽c⩽d⩽a$ with a>2$a>2$, we prove that any nonnegative integer can be written as x(ax+b)+y(ay+c)+z(az+d)$x(ax+b)+y(ay+c)+z(az+d)$ with x,y,z$x,y,z$ integers, if and only if the quadruple (a,b,c,d)$(a,b,c,d)$ is among