Related papers: On $p$-adic congruences involving $\sqrt d$

Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study…

In this paper, we consider sums of class numbers of the type $\sum_{m\equiv a\pmod{p}} H(4n-m^2)$, where $p$ is an odd prime, $n\in \mathbb{N},$ and $a\in \mathbb{Z}$. By showing that these are coefficients of mixed mock modular forms, we…

We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value…

For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we…

In this paper, we confirm some congruences conjectured by V.J.W. Guo and M.J. Schlosser recently. For example, we show that for primes $p>3$, $$…

Let $p$ be an odd prime and $r\geq 1$. Suppose that $\alpha$ is a $p$-adic integer with $\alpha\equiv2a\pmod p$ for some $1\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that…

In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed…

Let $p$ be a prime and let $x$ be a $p$-adic integer. We provide two supercongruences for truncated series of the form $$\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots…

Let $\sigma(n)$ and $\gamma(n)$ denote the sum of divisors and the product of distinct prime divisors of $n$ respectively. We shall show that, if $n\neq 1, 1782$ and $\sigma(n)=(\gamma(n))^2$, then there exist odd (not necessarily distinct)…

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd}…

We prove, that the sequence $1!, 2!, 3!, \dots$ produces at least $(\sqrt{2} + o(1))\sqrt{p}$ distinct residues modulo prime $p$. Moreover, factorials on an interval $\mathcal{I} \subseteq \{0, 1, \dots, p - 1\}$ of length $N > p^{7/8 +…

Let $n\geq 3$ be an integer and $p$ be a prime with $p\equiv 1\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\bigg[\begin{matrix} \frac{n-1}{n}&\frac{n-1}{n}&\ldots&\frac{n-1}{n}\\ &1&\ldots&1\end{matrix}\bigg | \, 1\bigg]_{p-1}\equiv…

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case…

Let $t\in\mathbb{N}_+$ be given. In this article we are interested in characterizing those $d\in\mathbb{N}_+$ such that the congruence $$\frac{1}{t}\sum_{s=0}^{t-1}{n+d\zeta_t^s\choose d-1}\equiv {n\choose d-1}\pmod{d}$$ is true for each…

Let $m\in\N_{\geq 2}$ and for given $k\in\N_{+}$ consider the sequence $(A_{m,k}(n))_{n\in\N}$ defined by the power series expansion $$ \prod_{n=0}^{\infty}\frac{1}{\left(1-x^{m^{n}}\right)^{k}}=\sum_{n=0}^{\infty}A_{m,k}(n)x^{n}. $$ The…

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by…

The central Delannoy numbers $D_n=\sum_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}$ and the little Schr\"oder number $s_n=\sum_{k=1}^{n}\frac{1}{n}\binom{n}{k}\binom{n}{k-1}2^{n-k}$ are important quantities. In this paper, we confirm…

We present a formal verification of Wolstenholme's theorem -- $\binom{2p}{p} \equiv 2 \pmod{p^3}$ for prime $p \geq 5$ -- in Lean~4 with Mathlib. The proof proceeds by expanding the shifted factorial product $\prod_{k=1}^{p-1}(p+k)$ to…

Let $p_n$ denote the $n$-th prime number, and let $d_n=p_{n+1}-p_{n}$. Under the Hardy--Littlewood prime-pair conjecture, we prove \begin{align*} \sum_{n\le X}\frac{\log^{\alpha}d_n}{d_n} \sim\begin{cases} \frac{X\log\log\log X}{\log…

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures…