Related papers: Congruences for sequences analogous to Euler numbe…

For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$…

Let $[x]$ be the greatest integer not exceeding $x$. In the paper we introduce the sequence $\{U_n\}$ given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]}\binom n{2k}U_{n-2k}\quad(n\ge 1)$, and establish many recursive formulas and congruences…

Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0\pmod p$. In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k…

We prove that for a positive integer a the integer sequence P(n) satisfying for all n, -infty<n<infty, the recurrence P(n)=a+P(n-phi(a)), phi(a) the Euler function, generates in increasing order all integers P(n) coprime to a.The finite…

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$…

We present a theorem which generalizes the classical Euler's theorem on congruencies: if $(a,m)=1$ then $a^ \phi(m) \equiv 1 (mod m)$ for the case when $a$ and $m$ are not relatively primes.

We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer…

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q),…

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum_{k=0}^{p-1}\binom…

For $n=0,1,2,\ldots$ let $W_n=\sum_{k=0}^{[n/3]}\binom{2k}k \binom{3k}k\binom n{3k}(-3)^{n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then $\{W_n\}$ is an Ap\'ery-like sequence. In this paper we deduce many congruences…

For m>n\geq 0 and 1\leq d\leq m, it is shown that the q-Euler number E_{2m}(q) is congruent to q^{m-n}E_{2n}(q) mod (1+q^d) if and only if m\equiv n mod d. The q-Sali\'e number S_{2n}(q) is shown to be divisible by…

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

Let $(a;q)_n=\prod_{0\le k<n}(1-aq^k)$ for n=0,1,2,.... Define q-Euler numbers $E_n(q)$, q-Sali\'e numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows: $$\sum_{n=0}^{\infty}E_n(q)\frac{x^n}{(q,q)_n}…

The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of…

Let $p$ be an odd prime and let $x$ be a $p$-adic integer. In this paper, we establish supercongruences for $$ \sum_{k=0}^{p-1}\frac{\binom{x}{k}\binom{x+k}{k}(-4)^k}{(dk+1)\binom{2k}{k}}\pmod{p^2} $$ and $$…

In this article, we introduce congruential Euler numbers, which are a further generalization of generalized Euler numbers. We prove the $p$-adic congruences of congruential Euler numbers, which include answers to a conjecture related to…

Let $n=p_1^{\nu_1}... p_r^{\nu_r} >1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^2x\equiv a$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers $a$ (mod $n$) such that…

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…