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 formulatext stixSupport mathImg" data-mathURL="/science?_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,cflow="scroll">a,b,c every nonnegative integer n   can be written as formulatext stixSupport mathImg" data-mathURL="/science?_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)flow="scroll">xfalse">(ax+1false">)+yfalse">(by+1false">)+zfalse">(cz+1false">) with formulatext stixSupport mathImg" data-mathURL="/science?_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,zflow="scroll">x,y,z integers. We show that formulatext stixSupport mathImg" data-mathURL="/science?_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)flow="scroll">false">(a,b,cfalse">) can be either of the following seven triples and conjecture that any triple formulatext stixSupport mathImg" data-mathURL="/science?_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)flow="scroll">false">(a,b,cfalse">) among also has the desired property. For integers formulatext stixSupport mathImg" data-mathURL="/science?_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⩽aflow="scroll">0⩽b⩽c⩽d⩽a with formulatext stixSupport mathImg" data-mathURL="/science?_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>2flow="scroll">a>2, we prove that any nonnegative integer can be written as formulatext stixSupport mathImg" data-mathURL="/science?_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)flow="scroll">xfalse">(ax+bfalse">)+yfalse">(ay+cfalse">)+zfalse">(az+dfalse">) with formulatext stixSupport mathImg" data-mathURL="/science?_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,zflow="scroll">x,y,z integers, if and only if the quadruple formulatext stixSupport mathImg" data-mathURL="/science?_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)flow="scroll">false">(a,b,c,dfalse">) is among