On x(ax ce:hsp>+ ce:hsp>1)+y(by ce:hsp>+ ce:hsp>1)+z(cz ce:hsp>+ ce:hsp>1) and x(ax ce:hsp>+ ce:hsp>b)+y(ay ce:hsp>+ ce:hsp>c)+z(az ce:hsp>+ ce:hsp>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 ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=9027da55c12b0b6fb4d41dd79acb3a28" title="Click to view the MathML source">a,b,c$a,b,c$ every nonnegative integer n   can be written as ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=6a5001eddc312421f4aeffc4b195d9f9" title="Click to view the MathML source">x(ax+1)+y(by+1)+z(cz+1)$x(ax+1)+y(by+1)+z(cz+1)$ with ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=9d208e0acc3fbc1d4cc089861c24b469" title="Click to view the MathML source">x,y,z$x,y,z$ integers. We show that ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si35.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=b7813060b454ecb1bba6541779e11f80" title="Click to view the MathML source">(a,b,c)$(a,b,c)$ can be either of the following seven triples and conjecture that any triple ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si35.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=b7813060b454ecb1bba6541779e11f80" title="Click to view the MathML source">(a,b,c)$(a,b,c)$ among also has the desired property. For integers ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si7.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=27825d82117c9d93eb9f38b74b7ebd4f" title="Click to view the MathML source">0⩽b⩽c⩽d⩽a$0⩽b⩽c⩽d⩽a$ with ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si50.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=2dc6b613ec861a57b4a876bec1248663" title="Click to view the MathML source">a>2$a>2$, we prove that any nonnegative integer can be written as ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si9.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=4303567edbfa7881d0b30a7cdbedf05d" title="Click to view the MathML source">x(ax+b)+y(ay+c)+z(az+d)$x(ax+b)+y(ay+c)+z(az+d)$ with ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=9d208e0acc3fbc1d4cc089861c24b469" title="Click to view the MathML source">x,y,z$x,y,z$ integers, if and only if the quadruple ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si10.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=f0ae7e480986fc95477a4ab5e8fd53d7" title="Click to view the MathML source">(a,b,c,d)$(a,b,c,d)$ is among