Arithmetic properties of bipartitions with 3-cores

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• 作者：Ernest X. W. Xia
• 关键词：Bipartitions ; $$t$$ t ; cores ; Congruences ; 11P83 ; 05A17
• 刊名：The Ramanujan Journal
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：38
• 期：3
• 页码：529-548
• 全文大小：503 KB
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• 作者单位：Ernest X. W. Xia (1)
  
  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu, People鈥檚 Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Number Theory
  Field Theory and Polynomials
  Combinatorics
  Fourier Analysis
  Functions of a Complex Variable
  
• 出版者：Springer U.S.
• ISSN：1572-9303

文摘

Recently, Lin discovered several nice congruences modulo 4, 5, 7 and 8 for \(A_3(n)\), where \(A_3(n)\) is the number of bipartitions with 3-cores of \(n\). For example, Lin proved that for all \(\alpha \ge 0\) and \(n\ge 0\), \(A_3(16^{\alpha +1}n+\frac{2^{4\alpha +3}-2}{3})\equiv 0 \ (\mathrm{mod }\ 5)\). Yao also established several infinite families of congruences modulo 3 and 9 for \(A_9(n)\). In this paper, several infinite families of congruences modulo 4, 8 and \(\frac{4^k-1}{3}\ (k\ge 2)\) for \(A_3(n)\) are established. We generalize some results due to Lin and Yao. For example, we prove that for \(n\ge 0, \ \alpha \ge 0\) and \(k\ge 2\), \( A_3\left( 4^{k(\alpha +1)} n+\frac{2^{2k(\alpha +1)-1}-2}{3}\right) \equiv 0 \ (\mathrm{mod\ } \frac{4^{k}-1}{3}) \). Keywords Bipartitions \(t\)-cores Congruences