Powers of two weighted sum of the first p divided Bernoulli numbers modulo p

Congruences related to Miki's identity

Given an odd prime p, we present three independent ways of relating modulo p certain truncated convolutions of divided Bernoulli numbers to certain full convolutions of divided Bernoulli numbers.

Wilson's theorem modulo p^2 derived from Faulhaber polynomials

First, we present a new proof of Glaisher's formula dating from 1900 and concerning Wilson's theorem modulo p^2. Our proof uses p-adic numbers and Faulhaber's formula for the sums of powers (17th century), as well as more recent results on Faulhaber's coefficients obtained by Gessel and Viennot. Second, by using our method, we find a simpler proof than Sun's proof regarding a formula for (p-1)! modulo p^3, and one that can be generalized to higher powers of p. Third, we can derive from our method a way to compute the Stirling numbers modulo p^3, thus improving Glaisher and Sun's own results fr…

A congruence concerning a convolution involving weighted Bernoulli numbers

Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.

New super congruences involving Bernoulli and Euler polynomials

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 $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k \quad\text{and}\quad\sum_{k=0}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}{2k-1}\pmod {p^2}$$ in terms of Bernoulli and Euler polynomials. We also give some transformation formulas for congruences modulo $p^2$.

Sun的两个Legendre符号行列式与Bernoulli数

Two Legendre-symbol determinants of Sun and Bernoulli numbers

本文给出Sun定义的两个Legendre符号行列式模p的显式公式（用Bernoulli数表示），并利用Reinhart的反例找到使行列式模p为零的素数，否定回答了Sun的一个问题。

Fleck quotients and Bernoulli numbers

Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\sum_{k=r(mod p)}\binom{n}{k}(-1)^k\in\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m\not=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $0<n_*<p-2$ then $$\frac{F_p(pn,0)}{pn}=\frac{n_*…