Congruences involving alternating harmonic sums modulo $p^αq^β$

Rok, strany: 2018, 975 - 980

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} ∑_{\substack{i+j+k=p^r\i,j,k\in \mathcal{P}_p}}((1) / (ijk))\equiv-2p^r-1 B_p-3 \pmod{p^r}, \end{equation*} where $\mathcal{P}_n$ denote the set of positive integers which are prime to $n$. In this note, we obtain the congruences for distinct odd primes $p, q$ and positive integers $α, β$, \begin{equation*} ∑_{\substack{i+j+k=p^αq^β\i,j,k\in\mathcal{P}_2pq}}((1) / (ijk))\equiv ((7) / (8))(2-q)(1-((1) / (q³)))p^α-1q^β-1B_p-3\pmod{p^α} \end{equation*} and \begin{equation*} ∑_{\substack{i+j+k=p^αq^β\i,j,k\in \mathcal{P}_pq}} (((-1)ⁱ) / (ijk)) \equiv ((1) / (2))(q-2)(1-((1) / (q³)))p^α-1q^β-1B_p-3 \pmod{p^α}.

ISO 690:

Shen, Z., Cai, T. 2018. Congruences involving alternating harmonic sums modulo $p^αq^β$. In Mathematica Slovaca, vol. 68, no.5, pp. 975-980. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0159

APA:

Shen, Z., Cai, T. (2018). Congruences involving alternating harmonic sums modulo $p^αq^β$. Mathematica Slovaca, 68(5), 975-980. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0159