Related papers: Variations of Lucas' Theorem Modulo Prime Powers

Let $p$ be a prime and let $g(p)$ be the least primitive root modulo $p$. We prove that for any $\epsilon>0$ and $p$ large enough the bound \begin{align} g(p)\ll p^{\frac{1}{4\sqrt{e}}+\epsilon} \nonumber \end{align} holds for most prime…

Let $\overline{p}_{k}(n)$ denote the number of overpartition $k$-tuples of $n$. In 2023, Saikia \cite{saikia} conjectured the following congruences: \begin{align*} \overline{p}_{q}(8n+2)& \equiv 0 \pmod{4},\quad \overline{p}_{q}(8n+3)\equiv…

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…

We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…

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

In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's…

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $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,…

Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p.…

Let $k\ge 1$ be an integer, and let $(U_n)$ be the Lucas sequence of the first kind defined by \begin{equation*}\label{Eq:Lucas} U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}. \end{equation*} It is…

Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every…

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

Andrews and Paule revisited combinatorial structures known as the $k$-elongated partition diamonds, which were introduced in connection with the study of the broken $k$-diamond partitions. They found the generating function for the number…

In the previous work, Lim and the author determined the rank variety of the simple $\mathbb{F}\mathfrak{S}_{kp}$-module $D(p-1)=D^{(kp-p+1,1^{p-1})}$ with respect to some maximal elementary abelian $p$-subgroup $E_k$ and the complexity when…

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…

Let $p$ be an odd prime. Define the Gaussian power sum \[ G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n\in\mathbb Z[i]. \] We determine $G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$G_p(p)\equiv p^2(1+i)\pmod{p^3},$$…

We present a deterministic algorithm that, given a prime $p$ and a solution $x \in \mathbb Z$ to the discrete logarithm problem $a^x \equiv b \pmod p$ with $p\nmid a$, efficiently lifts it to a solution modulo $p^k$, i.e., $a^x \equiv b…

In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Delannoy numbers $D_k$ and polynomials $D_k(z)$. Let $v\in\bN$ and $p$ be an odd prime. It is proved that, for any…

In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums…

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p, N) \in \{(3, 1), (5, 1), (7, 1), (3, 4)\}$. In these…

Let $p$ be an odd prime. It is well known that $F_{p-(\frac p5)}\equiv 0\pmod{p}$, where $\{F_n\}_{n\ge0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\not=5$ then we may determine $F_{p-(\frac…