Note on some congruences of Lehmer

Corrigendum to "Cullen numbers with the Lehmer Property"

In this note, we correct an oversight from the paper mentioned in the title.

迈向adelic曲线上Lehmer问题的推广

Toward a generalization of Lehmer's problem to adelic curves

本文探讨了Lehmer问题在adelic曲线上的推广，研究其在有限生成字段中的应用。

A note on the Voronoi congruences and the residue of the Fermat quotient

Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruence…

椭圆曲线 Lehmer 猜想的实验研究

Experimental investigations on Lehmer's conjecture for elliptic curves

本文提出计算椭圆曲线上固定次数数域中最小典范高度非挠点的方法，并基于收集的数据实验研究 Lehmer 与 Lang 猜想。

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer $n$ has the Lehmer property if $ϕ(n)$ divides $n-1,$ where $ϕ$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set $$ \left\{\frac{g^{n}-1}{g-1}\ \bigg|\ n,g\in\mathbb{N},\ ν_{2}(g)+ν_{2}(g+1)\leq L\ \right\}, $$ where $ν_{2}(g)$ denotes the highest power of $2$ that divides $g$, and $L\geq 1$ is a fixed real n…

广义欧拉数的 $p$-adic 同余及其与偶 Zeta 值的关系

$p$-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values

本文从 $p$-adic 和分析角度研究广义欧拉数，引入同余欧拉数并建立其 $p$-adic 同余，包括解决与 Lehmer 数相关的猜想，并利用复分析方法用同余欧拉数表示偶 Zeta 值。

Some implications of the Gessel identity

We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), Ernvall-Metsankyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2001), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsankyla residues to some full convolutions of the same kind. We also establish some congruences concerning other related weighed sums of powers of …

线性同余与 Bibak 的一个猜想

Linear Congruences and a Conjecture of Bibak

本文研究线性同余方程的计数问题，推广 Bibak、Lehmer 和 Ramanathan 的结果，给出平方剩余解及有序、严格有序解在特殊情形下的显式计数式。

A note on arithmetic congruences

By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.

Notes on congruence zeta functions via a Berkovich approach

Notes on congruence zeta functions via a Berkovich approach

本文在Scholze的Berkovich动机框架下，利用范畴迹构造了一种新的zeta函数，并证明其与经典同余zeta函数一致。

带同余限制的多元线性同余方程组

Linear Congruences in several variables with congruence restrictions

本文研究带限制条件的多元线性同余方程组，利用 Ramanujan 和给出解数的充要条件与显式公式，并推广至 $\mathbb{F}_q[t]$ 函数域。

A note on congruence properties of the generalized bi-periodic Horadam sequence

In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1 + cw_n-2; if n is odd with arbitrary initial conditions w_0, w_1 and nonzero real numbers a, b, and c. We investigate some congruence properties of the generalized Horadam sequence {w_n}.

A generalisation of a congruence due to Vantieghem only holding for primes

In this note we present a family of congruences which hold if and only if a natural number $n$ is prime.

关于一般同余子群上艾森斯坦级数空间的注记

A note on the spaces of Eisenstein series on general congruence subgroups

本文提出利用主同余子群的Hecke艾森斯坦级数理论研究一般同余子群上谱艾森斯坦级数的新方法，得到相关空间的基与系数性质，并简化证明了Eichler-Shimura同构定理。

Note on Igusa's cusp form of weight 35

A congruence relation satisfied by Igusa's cusp form of weight 35 is presented. As a tool to confirm the congruence relation, a Sturm-type theorem for the case of odd-weight Siegel modular forms of degree 2 is included.

Congruence relations of GSpin Shimura varieties

In these notes I proved the Chai-Faltings version of Eichler-Shimura congruence relation for simple GSpin Shimura varieties. This extends the results by Bueltel, Wedhorn and Koskivirta.

三个Brillhart-Lehmer-Selfridge素性证明用于Wagstaff数

Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers

本文基于Brillhart-Lehmer-Selfridge (BLS) N-1准则，对Wagstaff数W_2617、W_10501和W_12391进行了严格验证的素性证明，不依赖ECPP，仅使用经典N-1方法，并利用Cunningham项目表和FactorDB进行因子分解。

Notes on q-analogues of a binomial congruence of Glaisher

Andrews once gave $q$-analogues of a binomial congruence of Glaisher, and he suggested perfect $q$-analogues. In this note we give ones meeting the demand of Andrews.

A Note on Congruences for Weakly Holomorphic Modular Forms

Let $O_L$ be the ring of integers of a number field $L$. Write $q = e^{2 πi z}$, and suppose that $$f(z) = \sum_{n \gg - \infty}^{\infty} a_f(n) q^n \in M_{k}^{!}(\operatorname{SL}_2(\mathbb{Z})) \cap O_L[[q]]$$ is a weakly holomorphic modular form of even weight $k \leq 2$. We answer a question of Ono by showing that if $p \geq 5$ is prime and $ 2-k = r(p-1) + 2 p^t$ for some $r \geq 0$ and $t > 0$, then $a_f(p^t) \equiv 0 \pmod p$. For $p = 2,3,$ we show the same result, under the condition that $2 - k - 2 p^t$ is even and at least $4$. This represents the "missing case" of a theorem proved …

Congruences betwen Hilbert modular forms: constructing ordinary lifts II

In this note we improve on the results of our earlier paper[BLGG12], proving a near-optimal theorem on the existence of ordinary lifts of a mod l Hilbert modular form for any odd prime l.

Jacobsthal 和 Jacobsthal-Lucas 数的 Lehmer 性质

Jacobsthal and Jacobsthal-Lucas numbers with Lehmar property

本文证明了 Jacobsthal 和 Jacobsthal-Lucas 数列中不存在具有 Lehmer 性质的数。

Notes on theta series for Niemeier lattices II

Following the paper "Note on theta series for Niemeier lattices", we study some congruence properties satisfied by the theta series associated with Niemeier lattices.

Notes on theta series for Niemeier lattices

Some explicit expressions are given for the theta series of Niemeier lattices. As an application, we present some of their congruence relations.

Belyi parametrisations of elliptic curves and congruence defects

In this note we consider questions about parametrisations of elliptic curves defined over number fields by quotients of the upper half-plane by finite index subgroups of SL_2(Z). We ask if we can choose such a parametrisation of an elliptic curve to reflect the field over which it is defined: this can be viewed as a generalisation of the Shimura-Taniyama-Weil conjecture.

A note on traces of singular moduli

We will generalize Osburn's work about a congruence for traces defined in terms of Hauptmodul associated to certain genus zero groups of higher levels.

Note on super congruences modulo $p^2$

Let $p$ be an odd prime, and let $m$ be an integer with $p\nmid m$. In this paper show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak\binom{-1-a}k}{m^k} \equiv 0\pmod p \quad\hbox{implies}\quad\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak \binom{-1-a}k}{m^k}\equiv 0\pmod {p^2}.$$

关于广义三次分拆模素数同余的一个注记

A note on congruences for generalized cubic partitions modulo primes

本文用经典q级数方法重新证明广义三次分拆函数的两条已知同余式，并给出素数p不≡1 mod 8时的无穷同余族。

A note on the generalized q-Euler numbers(2)

In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.

Congruences for 1-shell totally symmetric plane partitions

Let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of weight $n$. Recently, Hirschhorn and Sellers, Yao, and Xia established a number of congruences modulo 2 and 5, 4 and 8, and 25 for $f(n)$, respectively. In this note, we shall prove several new congruences modulo 125 and 11 by using some results of modular forms. For example, for all $n\ge 0$, we have \begin{align*} f(1250n+125)&\equiv 0 \pmod{125},\\ f(1250n+1125)&\equiv 0 \pmod{125},\\ f(2750n+825)&\equiv 0 \pmod{11},\\ f(2750n+1925)&\equiv 0 \pmod{11}. \end{align*}

Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums

Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.

Some Remarks on Small Values of $τ(n)$

A natural variant of Lehmer's conjecture that the Ramanujan $τ$-function never vanishes asks whether, for any given integer $α$, there exist any $n \in \mathbb{Z}^+$ such that $τ(n) = α$. A series of recent papers excludes many integers as possible values of the $τ$-function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $τ(n)$. We synthesize these results and methods to prove that if $0 < |α| < 100$ and $α\notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$, then $τ(n) \neq α$ for all $n > 1$. Moreover, if $α\in T$ and $τ(n) =…

Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/k$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$ \sum_{k=1}^{p-1}\frac{H_k^2}{k^2} \equiv4/5pB_{p-5}\pmod{p^2}, $$ which confirms the conjecture recently proposed by Z. W. Sun. Furthermore, we also prove two similar congruences modulo $p^2$.

An Extension of a Congruence by Tauraso

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/n$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k^2}\equiv \sum_{k=1}^{p-1}\frac{H_k^2}{k} \equiv\frac{3}{2p}\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}. $$ Notice that the first part of this congruence is recently proposed by R. Tauraso as a problem in Amer. Math. Monthly. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers and a combinatorial i…

7和11模下的某些受限划分函数的同余关系

A note on congruences modulo 7 and 11 for two restricted partition functions

本文通过q级数技术研究了某些受限划分函数在7和11模下的同余关系，并推广了之前使用模形式方法得到的结果。

Congruences for the Almkvist-Zudilin numbers

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$; while the latter (essentially) focuses on the maximal powers $r$ and $t$ such that $c(p^rn)$ is congruent to $c(p^{r-1}n)$ modulo $p^t$. This is called supercongruence. In this note, we prove modest supercongruences for certain sequences that have come to be known as the Almkvist-Zudilin numbers and two other naturally related ones.

关于某些 Rogers-Ramanujan 型恒等式系数的一些新同余及分拆理论解释

Some New Congruences and Partition-Theoretic Interpretations for the Coefficients of Some Rogers-Ramanujan Type Identities

本文利用正整数的 overpartition 和 colour partition，为某些 Rogers-Ramanujan 型恒等式系数提供了分拆理论解释，并证明了模 2 的幂的无限族同余。

A note on congruences for the difference between even cranks and odd cranks

Recently, Amdeberhan and Merca proved some arithmetic properties of the crank parity function $C(n)$ defined as the difference between the number of partitions of $n$ with even cranks and those with odd cranks and the sequence $a(n)$ whose generating function is the reciprocal of that of $C(n)$. The function $C(n)$ was first studied by Choi, Kang, and Lovejoy. In this note, we give new elementary proofs of some of their main results and extend them. In particular, we establish Ramanujan-type congruences modulo $5$ and $25$ for certain finite sums involving $C(n)$ and $a(n)$. Our proofs employ …

关于Hirschhorn和Sellers广义颜色划分的更多同余关系

Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers

本文研究了广义颜色划分数$a_k(n)$的同余性质，通过生成函数和θ函数等方法，发现了一些模2的幂和模11的同余关系。

A Note on the least squarefree number in an arithmetic progression

We prove an asymptotic formula for squarefree in arithmetic progressions with squarefree moduli, improving previous results by Prachar. The main tool is an estimate for counting solutions of a congruence inside a box that goes beyond what can be obtained by using the Weil bound.

Congruences on the Bell polynomials and the derangement polynomials

In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.

Two congruences involving Andrews-Paule's broken 3-diamond partitions and 5-diamond partitions

In this note, we will give proofs of two congruences involving broken 3-diamond partitions and broken 5-diamond partitions which were conjectured by Peter Paule and Silviu Radu.

Congruences on the Number of Restricted $m$-ary Partitions

Andrews, Brietzke, Rødseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$ and thus confirm the conjecture of Andrews, et al.

Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. Work on Ramanujan--like congruence properties satisfied by the functions $cϕ_k(n)$ continues, with recent works completed b…

关于奇数分拆数的一个注记

A note on odd partition numbers

本文针对素数 $\ell\geq 5$，利用第二作者与 Ramsey 推广的 Mazur 结果及改进的 Sturm 准则，给出存在无穷多个 $m$ 使 $p(\ell m+δ_{\ell})$ 为奇数的新证明，并将最小 $m$ 的界改进至 $m<(\ell^2-1)/24$。

关于超几何数据$\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$及椭圆曲线对称平方的注记

A note on the hypergeometric datum $\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$ and symmetric squares of elliptic curves

本文探讨了超几何数据与椭圆曲线对称平方之间的模p同余关系。

以$0$结尾的迭代幂次基的同余速度

Congruence speed of tetration bases ending with $0$

本文针对以$0$结尾（即$10$的倍数）的迭代幂次基，推导出其在任意给定超指数高度下的同余速度公式，填补了该特定情形下同余速度计算的空白。

Another Smallest Part Function related to Andrews' spt function

In this note, we offer some relations and congruences for an interesting $spt$-type function.

关于4色广义Frobenius分拆的一个模5的意外同余

An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions

本文基于前人结果与Ramanujan的工作，用初等方法证明了4色广义Frobenius分拆函数$cϕ_4$满足一个模5的意外同余性质，补充了该函数的同余结论。

Proofs of Some Conjectures of Chan on Appell-Lerch Sums

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$ϕ(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=ϕ(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of $q$-products. As corollaries, we deduce the congruences $a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv0~(\textup{mod}~25)$ as well as $a(1250n+250r+219)\equiv 0~(\textup{mod}~125)$, where $r=1$, $3$, and $4$. The first three congruences were conjectured …

The number of cubic partitions modulo powers of 5

The notion of cubic partitions is introduced by Hei-Chi Chan and named by Byungchan Kim in connection with Ramanujan's cubic continued fractions. Chan proved that cubic partition function has Ramanujan Type congruences modulo powers of $3$. In a recent paper, William Y.C. Chen and Bernard L.S. Lin studied the congruent property of the cubic partition function modulo $5$. In this note, we give Ramanujan type congruences for cubic partition function modulo powers of $5$.

Congruences for Fishburn numbers modulo prime powers

The Fishburn numbers $ξ(n)$ are defined by the formal power series \[ \sum_{n \geq 0} ξ(n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j). \] Recently, G. Andrews and J. Sellers discovered congruences of the form $ξ(p m + j) \equiv 0$ modulo $p$, valid for all $m \geq 0$. These congruences have then been complemented and generalized to the case of $r$-Fishburn numbers by F. Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend …

A short note on sign changes

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group $SL_2(\mathbb{Z})$. We also prove a similar kind of quantitative result for the number of sign changes of the $q$-exponents $c(p) (p {vary over primes})$ of certain generalized modular functions for the congruence subgroup $Γ_0(N)$, where $N$ is square-free.

SL(3, Z)\SL(3, R) 中的交分布与三次同余根相关的格点

The distribution of intersections in $\mathrm{SL}(3, \mathbb{Z}) \backslash \mathrm{SL}(3, \mathbb{R})$ and lattices related to roots of cubic congruences

研究正对角子群闭轨与极大抛物子群平移的交点分布，证明交点在子群中联合等分布，并导出三次同余根对确定的仿射格点与理想联合等分布。

超立方分拆三元组的进一步算术性质

Further Arithmetic Properties of Overcubic Partition Triples

本文利用初等技巧和模形式理论，证明了超立方分拆三元组的新同余式，并推广至$k$-元组，扩展了2024年的已知结果。