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 class="mathmlsrc">class="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,cclass="mathContainer hidden">class="mathCode">croll">a,b,c every nonnegative integer n   can be written as class="mathmlsrc">class="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)class="mathContainer hidden">class="mathCode">croll">xchy="false">(ax+1chy="false">)+ychy="false">(by+1chy="false">)+zchy="false">(cz+1chy="false">) with class="mathmlsrc">class="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,zclass="mathContainer hidden">class="mathCode">croll">x,y,z integers. We show that class="mathmlsrc">class="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)class="mathContainer hidden">class="mathCode">croll">chy="false">(a,b,cchy="false">) can be either of the following seven triplesv class="formula" id="fm0010">v class="mathml">class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=e3be60a487642258bf2ab3ca3db24339">class="imgLazyJSB inlineImage" height="16" width="437" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302074-si5.gif">cript>vertical-align:bottom" width="437" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302074-si5.gif">cript>class="mathContainer hidden">class="mathCode">croll">chy="false">(1,2,3chy="false">),ce width="0.25em">ce>chy="false">(1,2,4chy="false">),ce width="0.25em">ce>chy="false">(1,2,5chy="false">),ce width="0.25em">ce>chy="false">(2,2,4chy="false">),ce width="0.25em">ce>chy="false">(2,2,5chy="false">),ce width="0.25em">ce>chy="false">(2,3,3chy="false">),ce width="0.25em">ce>chy="false">(2,3,4chy="false">),class="temp" src="/sd/blank.gif">v>v> and conjecture that any triple class="mathmlsrc">class="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)class="mathContainer hidden">class="mathCode">croll">chy="false">(a,b,cchy="false">) amongv class="formula" id="fm0020">v class="mathml">class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=45177d148d34c25e3dd003cd2ba10841">class="imgLazyJSB inlineImage" height="16" width="377" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302074-si6.gif">cript>vertical-align:bottom" width="377" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302074-si6.gif">cript>class="mathContainer hidden">class="mathCode">croll">chy="false">(2,2,6chy="false">),ce width="0.25em">ce>chy="false">(2,3,5chy="false">),ce width="0.25em">ce>chy="false">(2,3,7chy="false">),ce width="0.25em">ce>chy="false">(2,3,8chy="false">),ce width="0.25em">ce>chy="false">(2,3,9chy="false">),ce width="0.25em">ce>chy="false">(2,3,10chy="false">)class="temp" src="/sd/blank.gif">v>v> also has the desired property. For integers class="mathmlsrc">class="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⩽aclass="mathContainer hidden">class="mathCode">croll">0⩽b⩽c⩽d⩽a with class="mathmlsrc">class="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>2class="mathContainer hidden">class="mathCode">croll">a>2, we prove that any nonnegative integer can be written as class="mathmlsrc">class="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)class="mathContainer hidden">class="mathCode">croll">xchy="false">(ax+bchy="false">)+ychy="false">(ay+cchy="false">)+zchy="false">(az+dchy="false">) with class="mathmlsrc">class="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,zclass="mathContainer hidden">class="mathCode">croll">x,y,z integers, if and only if the quadruple class="mathmlsrc">class="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)class="mathContainer hidden">class="mathCode">croll">chy="false">(a,b,c,dchy="false">) is amongv class="formula" id="fm0030">v class="mathml">class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302074&_mathId=si11.gif&_user=111111111&_pii=S0022314X16302074&_rdoc=1&_issn=0022314X&md5=7f0b76aefada4b25ae4cc9425c790968">class="imgLazyJSB inlineImage" height="16" width="385" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302074-si11.gif">cript>vertical-align:bottom" width="385" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302074-si11.gif">cript>class="mathContainer hidden">class="mathCode">croll">chy="false">(3,0,1,2chy="false">),ce width="0.25em">ce>chy="false">(3,1,1,2chy="false">),ce width="0.25em">ce>chy="false">(3,1,2,2chy="false">),ce width="0.25em">ce>chy="false">(3,1,2,3chy="false">),ce width="0.25em">ce>chy="false">(4,1,2,3chy="false">).class="temp" src="/sd/blank.gif">v>v>