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 bc38454756894159d40b308b15" title="Click to view the MathML source">p₁=a²+b²$p_1=a^2+b^2$ and bbf3484f2766ada70e8a32026a" title="Click to view the MathML source">p₂=a²+(b+h)²$p_2=a^2+{(b+h)}^2$, with e4e123e0294ca9fccf046f48e5445" title="Click to view the MathML source">a,b,h$a,b,h$ integers, such that |h|≤246$|h|\le 246$. We do this by viewing a Gaussian prime baa7dbc44e49c9fe67114f87" title="Click to view the MathML source">c+di$c+di$ as a lattice point 98ef71d4fabca099" title="Click to view the MathML source">(c,d)$(c,d)$ in R²${\mathbb{R}}^2$ and showing that there are infinitely many pairs of distinct Gaussian primes 98310edab8108e92d8edf7f51e38" title="Click to view the MathML source">(c₁,d₁)$(c_1,d_1)$ and 9841ad5a38d8a43448e3ade1631db8" title="Click to view the MathML source">(c₂,d₂)$(c_2,d_2)$ such that the Euclidean distance between them is bounded by 246. Our method, motivated by the work of Maynard bbr0100">[9] and the Polymath project bbr0160">[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.