Cooper and Lam's conjecture for generalized Bell ternary quadratic forms

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• 作者：Werner Hü ; rlimann ^{whurlimann@bluewin.ch" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11E25 ; 11D45
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：158
• 期：Complete
• 页码：23-32
• 全文大小：281 K

文摘

Bell's theorem determines the counting function of the ternary quadratic forms 2334&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002334&_rdoc=1&_issn=0022314X&md5=491f15612135b6e0dd51466d3e444c99" title="Click to view the MathML source">x²+by²+cz²$x^2+b{y}^2+c{z}^2$, with 2334&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002334&_rdoc=1&_issn=0022314X&md5=4faa26e6ad5a48bb4eeae30b5f1040b2" title="Click to view the MathML source">b,c∈{1,2,4,8}$b,c\in \{1,2,4,8\}$, in terms of the number 2334&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002334&_rdoc=1&_issn=0022314X&md5=398240ae7212cea9c81ce963f61d2f79" title="Click to view the MathML source">r₃(n)$r_3(n)$ of representations of n   as a sum of three squares. Based on it we verify Cooper and Lam's conjecture for them. This result includes two new cases so far left open. Additionally, we show that the forms 2334&_mathId=si4.gif&_user=111111111&_pii=S0022314X15002334&_rdoc=1&_issn=0022314X&md5=90f223ffc71f6af436663f612a90606b" title="Click to view the MathML source">(b,c)=(2,16)$(b,c)=(2,16)$ and 2334&_mathId=si5.gif&_user=111111111&_pii=S0022314X15002334&_rdoc=1&_issn=0022314X&md5=dbeb371c40e1391b4aa8a1a588ef52e4" title="Click to view the MathML source">(b,c)=(8,16)$(b,c)=(8,16)$ are generalized Bell forms in the sense that their counting functions depend only upon 2334&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002334&_rdoc=1&_issn=0022314X&md5=398240ae7212cea9c81ce963f61d2f79" title="Click to view the MathML source">r₃(n)$r_3(n)$. These forms satisfy Cooper and Lam's conjecture and solve two further open cases.