Bounded gaps between Gaussian primes

详细信息    查看全文

• 作者：Akshaa Vatwani ^{akshaa@mast.queensu.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11N36 ; 11R44
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：449-473
• 全文大小：470 K

文摘

We show that there are infinitely many distinct rational primes of the form hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si39.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=68cf2cbc38454756894159d40b308b15" title="Click to view the MathML source">p₁=a²+b²hContainer hidden">hCode">h altimg="si39.gif" overflow="scroll">w>pw>w>1w>=w>aw>w>2w>+w>bw>w>2w>h> and hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=c439dfbbf3484f2766ada70e8a32026a" title="Click to view the MathML source">p₂=a²+(b+h)²hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">w>pw>w>2w>=w>aw>w>2w>+w>hy="false">(b+hhy="false">)w>w>2w>h>, with hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=703e4e123e0294ca9fccf046f48e5445" title="Click to view the MathML source">a,b,hhContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">a,b,hh> integers, such that hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=8fc54fb7a8c940d66dcec16d6fdaebe8" title="Click to view the MathML source">|h|≤246hContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">hy="false">|hhy="false">|≤246h>. We do this by viewing a Gaussian prime hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si5.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=03ab1195baa7dbc44e49c9fe67114f87" title="Click to view the MathML source">c+dihContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">c+dih> as a lattice point hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si6.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=6d911f937b63ff2c98ef71d4fabca099" title="Click to view the MathML source">(c,d)hContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">hy="false">(c,dhy="false">)h> in hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si7.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=6cf7f59665ad0dc94858110c19b6b450" title="Click to view the MathML source">R²hContainer hidden">hCode">h altimg="si7.gif" overflow="scroll">w>hvariant="double-struck">Rw>w>2w>h> and showing that there are infinitely many pairs of distinct Gaussian primes hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si8.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=0bdd98310edab8108e92d8edf7f51e38" title="Click to view the MathML source">(c₁,d₁)hContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">hy="false">(w>cw>w>1w>,w>dw>w>1w>hy="false">)h> and hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301913&_mathId=si9.gif&_user=111111111&_pii=S0022314X16301913&_rdoc=1&_issn=0022314X&md5=9f9841ad5a38d8a43448e3ade1631db8" title="Click to view the MathML source">(c₂,d₂)hContainer hidden">hCode">h altimg="si9.gif" overflow="scroll">hy="false">(w>cw>w>2w>,w>dw>w>2w>hy="false">)h> such that the Euclidean distance between them is bounded by 246. Our method, motivated by the work of Maynard [9] and the Polymath project [13], is applicable to the wider setting of imaginary quadratic fields with class number 1 and yields better results than those previously obtained for gaps between primes in the corresponding number rings.