Waring-Goldbach problem: Two squares and some higher powers

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• 作者：Yingjie Li^a ; Yingchun Cai^b ; ^{yingchuncai@mail.tongji.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11P32 ; 11N36
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：162
• 期：Complete
• 页码：116-136
• 全文大小：357 K

文摘

Let mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si1.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=c7ce8c96b23f14d6a7c914bfc74bfc4e" title="Click to view the MathML source">P_rmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">Prmath> denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every large odd integer N  , and mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si2.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=d106dafc53af232bc9d341e2dfb1cc61" title="Click to view the MathML source">5≤a≤8mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">5≤a≤8math>, mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si22.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=f63742a8d8fdd99b550a0d2799f56c63" title="Click to view the MathML source">a≤bmathContainer hidden">mathCode"><math altimg="si22.gif" overflow="scroll">a≤bmath>, mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si23.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=ab235544fea5a7055c57209764ae5801">mage" height="20" width="110" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15003613-si23.gif">mathContainer hidden">mathCode"><math altimg="si23.gif" overflow="scroll">1360<1a+1b≤13math>, the equation is solvable with x   being an almost-prime mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si188.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=2fb927786954c5b2649c6ae8ab02e613" title="Click to view the MathML source">P_r(a,b)mathContainer hidden">mathCode"><math altimg="si188.gif" overflow="scroll">Pr(a,b)math> and the other variables primes, where mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si7.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=97608dd1a82996993ab996b17402cc06" title="Click to view the MathML source">r(a,b)mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll">r(a,b)math> is defined in the Theorem, in particular, mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15003613&_mathId=si8.gif&_user=111111111&_pii=S0022314X15003613&_rdoc=1&_issn=0022314X&md5=7cd087ad755c9796c44c7016bea7274d" title="Click to view the MathML source">r(6,7)=5mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll">r(6,7)=5math>. This result constitutes an refinement upon that of J. Brűdern.