A congruence involving harmonic sums modulo $p^{\alpha }{q}^{\beta }$

Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} Z(p^{r})\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $Z(n)=\sum _{\genfrac{}{}{0ex}{}{i+j+k=n}{i,j,k\in {\mathcal{P}}_n}}\frac{1}{ijk}$ and ${\mathcal{P}}_n$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,q$ and positive integers $\alpha ,\beta$, \begin{equation*} Z(p^αq^β)\equiv 2(2-q)(1-\frac{1}{q^{3}})p^{α-1}q^{β-1}B_{p-3}\pmod{p^α} \end{equation*} and the necessary and sufficient condition for \begin{equation*} Z(p^αq^β)\equiv 0\pmod{p^αq^β}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^{\alpha }||n$, $\alpha \ge 1$, \begin{eqnarray} \nonumber Z(n)\equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{2n}{p})B_{p-3}\pmod{p^α}. \end{eqnarray}

References in corpus (1)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

Cited by in corpus (4)

• New Congruences on Multiple Harmonic Sums and Bernoulli Numbers
• A congruence involving alternating harmonic sums modulo $p^{\alpha }{q}^{\beta }$
• On the third largest prime divisor of an odd perfect number
• Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums