In this paper, we mainly prove some conjectural congruences of Z.-H. Sun involving Almkvist–Zudilin numbers <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11464_2024_0111_Article_Equ1.gif" Format="GIF" Height="59" Rendition="HTML" Resolution="72" Type="Linedraw" Width="338" /> </MediaObject> <EquationSource Format="TEX">${b_n} = \sum\limits_{k = 0}^{\left\lfloor {{n \over 3}} \right\rfloor} {\left({\matrix{{2k} \cr k \cr}} \right)\left({\matrix{{3k} \cr k \cr}} \right)} \left({\matrix{n \cr {3k} \cr}} \right)\left({\matrix{{n + k} \cr k \cr}} \right){\left({- 3} \right)^{n - 3k}}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>b</mi> <mi>n</mi> </msub> </mrow> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mrow> <mo>⌊</mo> <mrow> <mrow> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>⌋</mo> </mrow> </mrow> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mn>3</mn> <mi>k</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mtable> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>3</mn> <mi>k</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mi>k</mi> </mrow> </msup> </mrow> <mo>.</mo> </math></EquationSource> </Equation>Let p > 3 be a prime. If p ≡ 3 (mod 4), then

Proof of Some Congruence Conjectures of Z.-H. Sun

期刊论文
发表日期：

2025年1月5日

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Guoshuai Mao,
Zhengkai Zhao

摘要

In this paper, we mainly prove some conjectural congruences of Z.-H. Sun involving Almkvist–Zudilin numbers ${b_n} = \sum\limits_{k = 0}^{\left\lfloor {{n \over 3}} \right\rfloor} {\left({\matrix{{2k} \cr k \cr}} \right)\left({\matrix{{3k} \cr k \cr}} \right)} \left({\matrix{n \cr {3k} \cr}} \right)\left({\matrix{{n + k} \cr k \cr}} \right){\left({- 3} \right)^{n - 3k}}.$ $b_{n} = \sum_{k = 0}^{⌊ \frac{n}{3} ⌋} (\begin{matrix} 2 k \\ k \end{matrix}) (\begin{matrix} 3 k \\ k \end{matrix}) (\begin{matrix} n \\ 3 k \end{matrix}) (\begin{matrix} n + k \\ k \end{matrix}) {(- 3)}^{n - 3 k} .$

Let p > 3 be a prime. If p ≡ 3 (mod 4), then