相关论文: On the special harmonic numbers $H_{\lfloor p/9 \r…

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence: $$\sum_{0<i_1<...<i_n<p}(i_1/3)(-1)^{i_1}/(i_1...i_n)=0 (mod p).$$

We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on…

In [14] Hemmer conjectures that the module of fixed points for the symmetric group $\Sigma_m$ of a Specht module for $\Sigma_n$ (with $n>m$), over a field of positive characteristic $p$, has a Specht series, when viewed as a…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

By finding the congruent relations between the generating function of the 5 dots bracelet partitions and that of the 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a…

Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3\equiv (-1)^{(p-1)/2}p\pmod{p^3}.$$ In this paper we show further that…

In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $\delta\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/\delta}…

The celebrated Touchard congruence states that $B_{n+p}\=B_n+B_{n+1}$ modulo $p$, where $p$ is a prime number and $B_n$ denotes the Bell number. In this paper we study divisibility properties of $B_{n-p}$ and their generalizations involving…

We completely calculate the $RO(\mathbb{Z}/p)$-graded coefficients $H\underline{\mathbb{Z}/p}_\star H\underline{\mathbb{Z}/p}$ for the constant Mackey functor $\underline{\mathbb{Z}/p}$.

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan--Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime $p$. The conjectures…

Difficulty of calculation of discrete logarithm for any arbitrary Field is the basis for security of several popular cryptographic solutions. Pohlig-Hellman method is a popular choice to calculate discrete logarithm in finite field $F_p^*$.…

Let $p_{\{3, 3\}}(n)$ denote the number of $3$-regular partitions in three colours. In a very recent paper, da Silva and Sellers studied certain arithmetic properties of $p_{\{3, 3\}}(n)$. They further conjectured four Ramanujan-like…

Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1 is a…

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3…

Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain…

For a prime number $p>2$, we explain the construction of the difference divisors on the unitary Rapoport-Zink spaces of hyperspecial level and the GSpin Rapoport-Zink spaces of hyperspecial level associated to a minuscule cocharacter $\mu$…

In this paper, we employ the Wilf-Zeilberger (WZ) method to prove a supercongruence conjecture posed by Z.-W. Sun: for any prime $p$, \begin{align*} \sum_{k=0}^{\frac{p-3}{2}}\frac{92k^2+61k+9}{(2k+1)64^k}{2k \choose k}{3k \choose k}{4k…

The first part of the paper is a survey of recent results about the cohomology of $(\phi,\Gamma)$-modules and its applications to the theory of Selmer complexes. In the second part we formulate a version of the Main Conjecture for $p$-adic…

Let $p\in \mathbb{N}$ be prime, and let $\theta$ be irrational. The authors have previously shown that the noncommutative $p$-solenoid corresponding to the multiplier of the group $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$…