New harmonic number identities with applications

We determine the explicit formulas for the sum of products of homogeneous multiple harmonic sums $\sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{\lambda_j})$ when $\sum_{j=1}^r \lambda_j\leq 5$. We apply these identities to the study of two congruences modulo a power of a prime

Supercongruences for a truncated hypergeometric series

The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a}$$ for $a=1$ and $a=2$. In the last section, the special case $x=1/2$ is considered.Comment: Revised versio

Wolstenholme and Morley, Primes and Pseudoprimes

In this note, we prove $p^2$ is a Morley pseudoprime (of order 2) iff $p^2$ is a Wolstenholme pseudoprime (of order 2) iff $p$ is a Wolstenholme prime iff $p$ is a Morley prime. Concerning pseudoprimes of order 1 that are not powers of primes, only 3 are known of Wolstenholme's type and absolutely none have yet been identified of Morley's type

A Characterization of Wilson-Lerch Primes

This note presents criteria in terms of Bernoulli numbers for a prime to be simultaneously a Wilson prime and a Lerch prime.http://www.integers-ejcnt.org/q51/q51.Abstract.htm