Aidas Medžiūnas
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special cases.
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Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
同余发现模幂为5，7和13安德鲁斯的最小部分分拆函数spt（n）。这些同余让人想起拉马努金的分割同余模幂为5，7和11。最近，Ono证明了作者在前面已经证明的spt（n）模p对所有素数p>3的显式Ramanujan型同余。我们推广了Ono的方法来处理5，7和13的幂同余。对于戴德金η函数，我们需要弱Maass形式的理论以及某些经典的模方程。
Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for PD$_t(n)$ and PDO$_t(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for PD$_t(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also…
In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ for $B_{5,s}(n)$. Very recently, Kathiravan proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. In this paper, we will prove infinite families of congruences modulo $5$ for $B_{2,15}(n)$, modulo $11$ for $B_{7,11}(n)$, modulo $11$ for $B_{27,11}(n)$ and modulo $17$ for $B_…
Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to prove infinite families of congruences modulo $m$ for $(l,m)$-regular bipartitions, where $m\in\{7,3,11,13,17\}$.
We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.
Modulo 2 congruences for partitions with initial repetitions
本文针对带初始重复的分拆类，推导了相关函数的奇偶性公式，并提出了无穷多组模2拉马努金型同余式，丰富了分拆同余理论。
Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers
本文开始了对模形式和模素幂的模伽罗瓦表示的同余关系的计算研究。算法描述计算最大整数模，其中两个一元互质积分多项式有一个共同的根在某种意义上说，是定义。这些技巧被应用到研究的同余的模形式和模伽罗瓦表示模素幂。最后，一些计算结果的影响（非）提升的模形式模素幂和可能的推广的水平提高。
Recently, Nayaka and Naika (2022) proved several congruences modulo $16$ and $32$ for $t$-colored overpartitions with $t=5,7,11$ and $13$. We extend their list using an algorithmic technique.
Two partition functions with congruences modulo 3, 5, 7, and 13
我们引入了两个新的整数分拆函数，这两个函数都是具有一定大小限制的$n$的分割四元组的数目。利用广义Lambert级数恒等式和$q$ -级数技巧，证明了这两个函数满足模$3$，$5$，$7$和$13$的Ramanujan型同余.
Hecke-type congruences for Andrews' spt-function modulo 16 and 32
最近由Andersen与不同的权力，2在模的分区相关的功能的启发，我们扩展了模32760同余的第一作者的功能spt（n）。我们证明了spt（n）的生成函数的正规化形式是Hecke算子T（l^2）对素数l >= 5的模32的本征形，其中l同余于1，11，17，19（mod 24），以及对l同余于？13，23（mod 24）.
We establish the existence of congruences between a fixed endoscopic cuspidal automorphic representation $Π$ of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at $1$ of the adjoint $L$-function of $Π$ normalized by a suitable period.
The structure of Siegel modular forms modulo p and U(p) congruences
我们确定Siegel模形式的度g模一个素数p的环结构，延长Nagaoka的结果的情况下，度g=2。我们刻画了Jacobi形式和Siegel模形式的U（p）同余，并惊奇地发现了Siegel模形式在偶数和奇数次上的不同行为.
Congruences for Restricted Plane Overpartitions Modulo 4 and 8
2009年，Corteel、Savelief和Vuletić将超分割的概念推广到一个新的对象，称为平面超分割。在最近的工作中，作者考虑了一种被称为$k$ -rowed plane overpartition的限制形式，并证明了一种方法来获得这些和其他类型的组合生成函数的同余。本文利用其他技巧证明了几个限制和非限制的模$4$和$8$的平面过划分同余.
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite families of congruences for $b_l(n)$ modulo $l$, where $l=13,17,23$.
We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.
On the congruence ax+by = 1 modulo xy
给出了当n趋于无穷大时，丢番图方程（X+1/x）（Y+1/y）= n的解的个数的界.这些界限与形式ax+by = 1 modulo xy的同余的解的数目有关。
Lehmer's type congruences for lacunary harmonic sums
本文研究缺调和和的Lehmer型同余。
For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove some general congruences modulo $5$ and $7$ for the colour partition function $p_r(n)$ by considering some general values of $r$. To prove the congruences we employ some $q$-series identities which is also in the spirit of Ramanujan.
Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.
Let $NT(m, k, n)$ denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck's conjecture on congruences for $NT(m, k, n)$ modulo 5 and 7. Generalizing Andrews'results, Chern obtain congruences for $NT(m, k, n)$ modulo 11 and 13. More recently, the second author use the theory of Hecke operators to establish congruences for such partition statistics modulo powers of primes $\ell \ge 7$. In this paper, we obtain Andrews-Beck type congruences modulo powers of 5.
The partition function $ p_{[1^c11^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{11}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}.\] In this paper, we prove infinite families of congruences for the partition function $ p_{[1^c11^d]}(n)$ modulo powers of $11$ for any integers $c$ and $d$, which generalizes Atkin and Gordon's congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup $Γ_0(11)$.
Congruences with intervals and subgroups modulo a prime
我们得到了关于几乎所有模素数$p$的剩余都用一个小整数和$({\mathbb Z}/p{\mathbb Z})^*$的小乘法子群的一个元素的乘积表示的新结果。这些结果是基于J.Cilleruelo和M. Z. Garaev（2014年），
The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain algebraic and arithmetic characterizations for Jacobi sums of order 2l^2 .
New Ramanujan-type congruences for overpartitions modulo $11$ and $13$
本文通过模形式理论推导出11和13模下的新分拆恒等式，并推测其他模数的可能恒等式。
Let $b_{\ell, k}(n), b_{\ell, k, r}(n)$ count the number of $(\ell, k)$, $(\ell, k, r)$-regular partitions respectively. In this paper we shall derive infinite families of congruences for $b_{\ell, k}(n)$ modulo $2$ when $ (\ell, k) = (3,8), (4, 7)$, for $b_{\ell, k}(n)$ modulo $8$, modulo $9$ and modulo $12$ when $(\ell, k) = (4, 9)$ and $b_{\ell, k, r}(n)$ modulo $2$ when $(\ell, k, r) = (3, 5, 8)$.
Congruences for central binomial sums and finite polylogarithms
我们证明了同余，模幂的素数p，某些有限和涉及中心二项式系数$\binom{2k}{k}$。
Let $\overline{p}_{j,k}(n)$ denotes the number of $(j,k)$-regular overpartitions of a positive integer $n$ such that none of the parts is congruent to $j$ modulo $k$. Naika et. al. (2021) proved infinite families of congruences modulo powers of 2 for $\overline{p}_{3,6}(n)$, $\overline{p}_{5,10}(n)$ and $\overline{p}_{9,18}(n)$. In this paper, we obtain infinite families of congruences modulo power of 2 for $\overline{p}_{4,8}(n)$, $\overline{p}_{6,12}(n)$ and $\overline{p}_{8,16}(n)$. For example, we prove that, for all integers $n\geq 0$ and $α\geq 0$, $$ \overline{p}_{4,8}\left( 5^{2α+1}\le…
Another proof of two modulo 3 congruences and another SPT crank for the number of smallest parts in overpartitions with even smallest part
通过考虑过划分和剩余曲柄的$M_2$ -秩，我们给出了同余$\overline{\mbox{spt}}_2(3n)\equiv \overline{\mbox{spt}}_2(3n+1)\equiv 0\pmod{3}$的另一种组合加细.这里，$\overline{\mbox{spt}}_2(n)$是$n$的超分区中最小部分的出现总数，其中最小部分是偶数并且没有上划线。我们的证明依赖于Bailey引理和Lovejoy和Osburn关于过划分的$M_2$ -秩的秩差公式。这个同余，以及一个模$5$同余，以前已经使用过分区的秩进行了细化。
Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences
对Andrews的最小部分分拆函数spt（n）找到了新的同余式. spt（n）的生成函数与某个权为3/2的弱马斯形式M（z）的全纯部分α（24 z）有关。我们证明了对于素数p > 3，spt（n）的生成函数的正规化形式是Hecke算子T（p^2）模72的本征形，如果（t，6p）= 1，则对于t = 5，7或13，本征形模t。模3的结果是作者早些时候观察到的，并由Ono和Folsom考虑。对于t的更高次幂（即5^6、7^4和13^2），函数alpha（z）的系数也会出现类似的同余。类似的结果分拆函数发现阿特金在1966年。我们的结果依赖于Ono最近的结果：M[p]（z/24）是满模群的权为3/2的弱全纯模形式，其中M[p]（z）= M（z）|T（p^2）- chi（p）（1 + p）M（z）.
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
Generalized Legendre polynomials and related congruences modulo $p^2$
对于任意正整数$n$和变量$a$和$x$，我们定义广义勒让德多项式$P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$。设$p$是一个奇素数。本文证明了许多与$P_{p-1}(a,x)$有关的模$p^2$同余.例如，我们证明了$P_{p-1}(a,x)\e (-1)^{<a>_p}P_{p-1}(a,-x)\mod {p^2}$，其中$<a>_p$是$a$模$p$的最小非负残基。推广了孙志伟的一些同余式，确定了$\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k{54^{-k}}$和$\sum_{k=0}^{p-1}\binom ak\binom{b-a}k\mod {p^2}$，其中$[x]$是最大整数函数.最后给出了二元二次型模$p^2$的超同余.
Congruences for L-functions of additive exponential sums
给出了由有限域上仿射可加指数和构成的L函数的一个同余式。准确地说，我们给出了一个同余某些运营商来自Dwork的理论。这是非常相似的同余Manin的特点多项式的行动Frobenius的雅可比曲线定义在一个有限域。
On q-Analog of Wolstenholme Type Congruences for Multiple Harmonic Sums
多重调和和是调和和的迭代推广。最近Dilcher认为同余涉及q-类似物这些款项的深度之一。本文将利用多重调和和的q-模拟的生成函数和混洗关系，研究任意深度的齐次情形。最后，我们还考虑了一些非齐次情形。
New congruences for 4,6-regular partitions modulo primes
本文证明了 b_4(n) 和 b_6(n) 分别对素数 m≥2 和 m≥5 存在无穷多 Ramanujan 型同余式族，并给出新例子及两个 b_4(n) 模 3 的无穷显式同余式族。
Some congruences modulo 5 and 25 for overpartition
我们提出了两个新的Ramanujan型同余模5的过度划分。我们还肯定地回答了Dou和Lin的一个猜想，该猜想包含四个模25同余。
Congruences for the number of partitions and bipartitions with distinct even parts
令$ped(n)$表示$n$的分区的数目，其中偶数部分是不同的（并且奇数部分是不受限制的）。我们证明了$ped(n)$模$8$的无穷族同余。我们还详细研究了$ped_{-2}(n)$模$8$的行为，其中$ped_{-2}(n)$表示$n$的偶数部分不同的二分区的数量。其结果是，我们找到了无穷族的同余$ped_{2}(n)$模$8$。
Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$
本文针对 Lin 关于 $\text{PDO}_t(n)$ 的猜想，提出新方法研究其关于 $3$ 的幂和 $2$ 的幂的同余式，建立了无穷多个模 $8$、$32$ 的同余式并证明若干小幂次同余式。
For each of the $15$ known sporadic Apéry-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of $ζ(3)$. Moreover, we show that each of these sequences satisfies two-term supercongruences modulo $p^{2r}$. Using special constant term representations recently discovered by Gorodetsky, we prove these supercongruences in the two cases that remained previously open.
Some Congruences of Kloosterman Sums and their Minimal Polynomials
利用Stickelberger定理和Gross-Koblitz公式证明了有限域上Kloosterman和的两个结果.第一个结果涉及的极小多项式在Q上的Kloosterman和，和第二个结果给出了一个特征的三元Kloosterman和模27。
Double exponential sums and congruences with intervals and exponential functions modulo a prime
设$p$是一个大素数，$g$是任意乘阶$T$模$p$的整数.本文给出了双指数和$$ S=\sum_{n\in \mathcal{N}}\left|\sum_{m\in \mathcal{M} }e_p(an g^{m})\right|, \quad \gcd (a,p)=1, $$的一个新的估计，其中$\mathcal{N}$和$\mathcal{M}$是具有$|\mathcal{N}|=N$和$|\mathcal{M}|=M<T$元素的连续整数的区间.一个代表性的例子是主要结果的以下结果：如果$N=M\approx p^{1/3}$，则$|S|< N^{2-1/8 + o(1)}$。然后，我们应用我们的估计，以获得新的结果添加剂同余涉及区间和指数函数。
On a symmetric congruence and its applications
对于大整数$m,$，我们得到了某个模$m$的四元同余的解的个数的一个渐近公式，这里的变量属于模$m.$的剩余类的特殊集合。这个公式被应用于模$m$的剩余环中的乘法表问题的例外集的新信息和一个关于指数函数的双三角和的新界。
A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes
本文证明了在特定条件下，不同素数乘积模下的整数幂和满足一种同余关系。
Congruences modulo prime powers of Hecke eigenvalues in level $1$
我们继续研究强，弱，和$dc$ -弱本征形介绍陈，基明，和Wiese。完全确定了$1$模$128$的Hecke本征值的所有系统，表明有许多。这扩展了Hatada的结果，可以被认为是更一般的猜想的证据，由作者与Kiming和Wiese一起制定的有限性系统的Hecke特征值模素数权力在任何固定的水平。我们还讨论了系统的Hecke本征值的水平$1$模$9$，减少问题的有限性的一个单一的本征值。此外，我们回答了比较弱和$dc$ -弱本征形的问题，并提供了第一个已知的非弱$dc$ -弱本征形的例子。
We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2, 3, 4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where $p$ is a prime.
A congruence modulo $n^3$ involving two consecutive sums of powers and its applications
对于各种正整数$k$，第一个$k$正整数的$k$次幂的和，$S_k(n+1)=1^k+2^k+...+n^k$，必须是所有数学中最流行的一些和。本文证明了对于每个$k\ge 2 $ $ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\equiv \{{array}{ll} 0\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\,even\,\,or}\,\, n\,\, {\rm is\,\, odd} & {\rm or} \,\, n\equiv 0\,(\bmod{\,4}) \frac{n^3}{2}\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\, odd} &,\,{\rm and}\,\, n\equiv 2\,(\bmod{\,4}). {array}.$ $ The above congruence allows us to state an equivalent formulation of Giuga's conjecture. Moreover, we prove that the first above congruence is satisfied modulo $ n ^4 $ whenever $ n \ge 5 $ is a prime number …
Lucas congruences for the Apéry numbers modulo $p^2$
本文给出 Apéry 数整函数插值的泰勒系数公式，并证明对特定基 $p$ 数字的整数 $n$，$A(n)$ 满足模 $p^2$ 的 Lucas 同余式。
Congruences of alternating multiple harmonic sums
在arXiv：0905.3327的续篇中，我们继续研究多重调和和的交替形式的同余性质。与Bernoulli数和伯努利多项式起关键作用的多重调和和的研究不同，在交替设置中欧拉数和欧拉多项式也是必不可少的。
On some congruence with application to exponential sums
我们将研究同余式$x \equiv g^{(1/2)ω_g(2^n)} \bmod 2^n$的解，它依赖于整数$g$和$n$，其中$ω_g(2^n)$表示$g$模$2^n$的阶。此外，我们介绍了上述结果的一个应用研究的估计指数和。
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are indeterminates. In the former case we deal more generally with shifted binomial coefficients $\binom{3k+e}{k}$. Our method derives such congruences directly from closed forms for the corresponding series.
Congruences of Power Sums
以下幂和的同余式$S_n(p)$是众所周知的，并且有许多应用：$1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n, \end{cases}$其中$n\in{\mathbb N}$和$p$是素数。特别地，我们将这个同余推广到$p$是素数的任意幂的情形。我们还证明了序列$(S_n(m) \text{ mod } k )_{m \geq 1}$是周期的，并确定了它的周期。
Congruences between modular forms modulo prime powers
给定一个素数$p \ge 5$和一个系数模为$p^n$（对于某些$n \ge 1$）和大图像的抽象奇表示$ρ_n$，我们证明了当局部提升存在时（在某些技术条件下），$ρ_n$提升到特征$0$的存在性.此外，我们可以在除了素数之外的所有素数上选择升力的惯性类型（其中升力是斯坦伯格型）。我们把这个结果应用到模形式的领域，证明了模素幂的一个降阶定理，并给出了升阶的例子。特别地，我们的方法表明，给定一个没有复数乘法或内部扭曲的模本征形$f$，对于所有的素数$p$，但不超过1000，并为所有的正整数$n$，存在另一个本征形$g\neq f$，这是全等的$f$模$p^n$。
Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case
我们采用Phong，Stein和Sturm的思想以及Ikromov和Müller的思想，从连续设置到各种离散设置，得到了二元一般拟齐次多项式的指数和的精确界以及多项式同余的解的个数.这扩展了工作的Denef和Sperber，也Cluckers关于猜想的Igusa在二维设置不再需要多项式是非退化的牛顿图。
Hecke Relations for Eta Multipliers and Congruences of Higher-Order Smallest Parts Functions
本研究通过作用于Eisenstein-eta商族的Hecke算子推导恒等式，得到系数素数幂模同余式，应用于解决Garvan提出的高阶最小分部函数同余问题，还建立曲柄矩、秩矩与分拆函数的素数幂模联系，强化推广了已有结果。
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.
Modified Congruence Modulo $n$ with Half The Amount of Residues
我们定义了一个新的同余关系的集合上的整数，导致一组类似的乘法群的整数模$n$。它利用了模乘中几乎无处不在的对称性，并将剩余类的数量减半。使用它，我们能够给出一个优雅的描述，由于卡尔Schick的一些结果，其他人减少到著名的定理，从代数和数论。数论中的许多概念，如二次剩余和原根，都是同样适用的。它在研究奇素数的幂，特别是在研究由一对相关素数组成的半素数，例如一对孪生素数时，带来了显著的优势。Artin的原根猜想可以在新的上下文中表述。基于弦的三角多项式与新的同余关系的联系，使人们对$2\cos(2π/n)$的极小多项式及其与分圆多项式的关系有了新的认识.
Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo. Our proof uses Ni and Pan's technique and a simple $q$-congruence observed by Guo and Schlosser.
Congruences modulo arbitrary powers of $5$ and $7$ for Andrews and Paule's partition diamonds with $(n+1)$ copies of $n$
本文证明了 Andrews 和 Paule 提出的分拆函数 PDN1(n) 关于 5 和 7 的任意幂的同余式，解决了他们提出的开放问题，并类比了 Ramanujan 和 Watson 的经典同余式。
Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series
证明了$q$ -阶乘比的多重和的模分圆多项式的同余关系，从而推广了许多著名的$p$ -Lucas同余关系.这样的同余将各种经典生成级数与它们的$q$ -类似物联系起来。利用这一点，我们证明了一个传播现象：当这些生成系列是代数独立的，这也是他们的$q$ -类似物的情况。
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